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SCIENCIA ACTA XAVERIANA

An International Research Journal of Basic and Applied Sciences

SCIENCIA ACTA XAVERIANA (SAX) is a referred biannual

research Journal published in March and September by

St. Xavier’s College (Autonomous), Palayamkottai - 627 002.

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References : Citations within the text are given within square brackets like [5], [9, 11],

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order :

[1] G. Chartrand and P. Zang, Distance in Graphs – Taking the long view, AKCE.J. Graphs,

1 No. 1 (2004), 1-13.

[2] M. Johnson, J. Vallinayagam, V.S. Manickam and S. Seeni, Multiplication of rhinacanthus

nasutus through micro propagation, Phytomorphology, 52, No.4 (2002), 331-336.

[3] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge

University Press, 2000.

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1

Volume 5

No. 2

pp. 1-26

September 2014

Sciencia Acta XaverianaAn International Science JournalISSN. 0976-1152

Tests on Vermicomposts for their suitability to

vegetable (Abelmoschus esculentus) crops

G.Sumathi1, M.Kanchilakshmi2, Arockiam Thaddeus3, P.Chandrasekar4

, P.Porkodi5 and

J.Sureka6

1 Research Scholar2 Project fellow3 Associate professor

Jayaraj Annapackiam College for Women, Periyakulam,

Theni District - 625601.TamilNadu, India.4 Principal, Mano college, Nagampatti, Tuticorin District – 628 7185,6 Assistant professors, SACWC, Cumbum.* Corresponding authors

Email : [email protected] ; [email protected]

ABSTRACT : Vermicomposting is a low cost technology for processing or

treatment of organic wastes. It contains N, P, K and micronutrients in forms that

are readily taken up by plants. For the present study, six genus of earthworms

(Lampito mauritii (T1A) , Octochetona pattoni (T

1B), Priodocheta pellucida (T

1C),

Notoscolex palniensis (T1D), Lemnoscolex scutarius (T

1E), and Hoplochetella

stuarti (T1F) collected from different ecosystems in Theni District were tried to

raise vermicomposts. These six different (T1A-B-C-D-E-F) treatments

(vermicomposts) were applied in different dosages (100,200,300 gm) to Ladies

finger plant (Abelmoschus esculentus) to test their efficacy. Treatments along with

control (T1) plants (Ladies finger) were reared. Highest observations were recorded

in Notoscolex palniensis (T1D) on growth and yield in 100 gm of compost

application. Soil samples were collected before and after the cultivation of

vegetables (Abelmoschus esculentus). These were subjected to physico-chemical

2

analysis of N, P, K pH, EC and micronutrients. These values were compared

with the values of plant height, number of fruits, yield, and length of fruit to apply

the regression analysis; there is a significant changes in soil parameters after the

application of vermicompost compared with control.

Keywords : Notoscolex palniensis, Vermicompost, Abelmoschus esculentus,

N, P, K, and Yield.

Introduction

Earthworms are major components of the soil fauna in a wide variety of soils and climates and

are involved directly or indirectly in biodegradation, stabilization through humus formation and

various soil processes (Lavelle and Spain 2001). The disposal of wastes through the use of

earthworms also upgrades the value of original waste materials insitu and allows a final product

to be obtained free of chemical or biological pollutants (Divya 2001). Vermicomposting involves

bio-oxidation and stabilization of organic material through the interactions between earthworms

and microorganisms. Although microorganisms are mainly responsible for the biochemical

degradation of organic matter, earthworms play an important role in the process by fragmenting

and conditioning the substrate, increasing the surface area for growth of microorganisms, and

altering its biological activity (Dominguez 2004; Dominguez and Edwards 2004). Vermicompost

contains major and minor nutrients in plant- available forms, enzymes, vitamins and plant growth

hormones. It has a more beneficial impact on plants than normal compost (Gajalakshmi and

Abbasi, 2004). Earthworms have been described as a keystone species as their activity helps

regulate soil fertility, water infiltration and soil detachability in agro ecosystems (Lavelle and

Spain, 2001; Shipitalo and LeBayon, 2004). Casts deposited on the soil surface may also be

carried away in surface water runoff after rain, leading to increased on-site soil erosion and

perhaps also affecting soil properties downstream of the worm’s actual home (Shipitalo and

Le Bayon, 2004). Direct effect of plant species on soil organisms are caused by the plant’s

inputs of organic matter above and below the ground, while indirect effect of plants on biota

include shading, soil protection and uptake of water and nutrients by the root (Neher, 1999).

According to Chaudhuri et al. (2003), the quality and quantity of food material influences not

only the size but also the species composition, growth rate, fecundity of an earthworm population.

They reported differences in the rate of growth and reproduction of three vermicomposting

Tests on Vermicomposts for their suitability to vegetable (Abelmoschus esculentus) crops

3

species Perionyx excavates, Eudrilus Eugenia and Eisenia fetida in the Hevea leaf litters used

as vermiculture substrate. Earthworms play an important role in maintaining soil fertility,

ecosystem function, production and biodiversity conservation (Chaudhuri et al., 2012; Kavdir

and Ilay 2011). As global food production is already dependent on intensive agricultural

production and demands for food are likely to increase substantially, the future challenge is to

match demands for production with forms of soil management that are sensitive to maintaining

soil biodiversity(Giller et al., 1997).

In the present study, six genus of earthworm adults [Lampito mauritii (T1A), Octochetona

pattoni (T1B), Priodocheta pellucida (T

1C), Notoscolex palniensis (T

1D), Lemnoscolex scutarius

(T1E) and Hoplochetella stuarti (T

1F)] collected from various ecosystems of Theni District

were selected and inoculated in decompost of vegetable refuses T1 to identify the suitable

genus of earthworm for vermicomposting out of six. Their efficacy was tested with Ladies

finger plants (Abelmoschus esculentus) in different dosages (100, 200, 300 gm).

Materials and Methods

Experiments were conducted during 2012-2013 at a farm land of Periyakulam, Theni District,

Tamilndu, India, to study the application of vermicompost and thereby to investigate the effect

of different dosages of vermicompost on vegetables.

Preparation of vermicompost and Germination

From the 6 selected earthworms (Octochetona pattoni, Priodocheta pellucida, Notoscolex

palniensis, Lemnoscolex scutarius, Hoplochetella stuarti and Lampito mauritii) were chosen as

experimental animal. Pre-composts were prepared with leaf litters of Manilkara zapota (sapota)

and Spathodea campanulata (Nandi Flame) separately mixed with cow dung (3:1) at regular

interval of 7 days over the soil bed and were used later in the earthworms were allowed to

feed on and converted them into vermicompost. It was harvested every 45 days.

Seeds of A. esculentus (Ladies finger) was sowed in the plotted fields and germinated when

cotyledons project out through the surface of the soil. Seedlings were planted at a distance of

30 cm between two plants. Lady’s Finger (A. esculentus) plants were grown in plots and were

applied with different dosages (100g, 200g & 300gm) of vegetable vermicompost raised from

G.Sumathi, M.Kanchilakshmi, Arockiam Thaddeus, P.Chandrasekar, P.Porkodi and J.Sureka

4

leaf litter. All the necessary cultural practices and plant protection measures were followed

uniformly for all the treatments during the entire period of experimentation and were replicated

four times in a randomized complete block design.

Following treatments were organized

1. Lampito mauritii Vermicompost (T1A)

2. Octochetona pattoni Vermicompost (T1B)

3. Priodocheta pellucida Vermicompost (T1C)

4. Notoscolex palniensis Vermicompost (T1D)

5. Lemnoscolex scutarius Vermicompost (T1E)

6. Hoplochetella stuarti Vermicompost (T1F)

7. Control (Farm soil)

Plant Growth Parameters

Vegetable refuse (T1A-B-C-D-E- F) vermicompost at the dosage of 100gm, 200gm and

300gm. Time of flowering, Height of the plant (cm), length of the fruit (cm), weight of the fruit

(gm), number of fruits per plant, yield of fruit per plant (gm) were recorded on every 10th day

from the date of seed germination up to 90 days.

Physico-Chemical Analysis

Soil samples were collected before and after the harvest of vegetables. These were subjected

to physico-chemical analysis (pH, electrical conductivity, organic carbon, total Kjeldahl nitrogen)

[Jackson, 1958]. Soil and vermicompost pH were measured in deionised water (solids/solution

ratio of 1: 2.5) using pH meter. Electrical conductivity (EC) was measured in the effluent and

in a saturated solution extract of the vermicompost (Rhoades, et al., 1989). Organic carbon

was determined by the Walkley-Black method (Gaudette, et al., 1974). Methods of measuring

N, P, and K in soil by Diethen-triamin-penta-acetic acid (DTPA), extractable Zn, Fe, Cu and


5

Mn were determined in soil samples by atomic absorption spectroscopy (Perkin Elmer, type

3041, series 3000) (Lindsay and Norvell, 1978).

Statistical analysis

Correlation and Regression analysis (SPSS computer version 17.0) was used to evaluate the

relationship between plant growth parameters and soil physicochemical parameter.

Result and Discussion

In the present study experiments were conducted to assess the requirements of vermicompost

by vegetable crops in bioremediation process to fulfil the needs of the farmers during the

transfer of technology at large scale. Among the different dosages (100, 200 & 300 gm) of

vegetable refuse vermicompost (T1A-B-C-D-E-F) applied (Abelmoschus esculentus) in the

present investigation, there has been a significant improvement in the soil quality of plots amended

with vermicompost @ 100 gm per plant (Table 1, 2, 3 and Fig 1, 2, 3). This is in concordance

with the results of the work of Edwards et al., 2000 that application of compost like

vermicompost enhances physical and chemical characteristics of soil in Bhendi (Abelmoschus

esculentus) cultivation. Vermicompost as an organic input has been applied to grow vegetables

and other crops succesfully (Ismail, S.A., 2005). Application of composts like vermicompost

could contribute to increased availablity of food (Ouedraogo, et al., 2001). The overall

productivity of vegetable crop Ladies finger during the 5 months of the trial was significantly

greater in plots treated with vegetable refuse (T1) @ 100 gm per plant.

Plant Height and Number of Branches

All treatments showed significant increase of plant height when compared to control plant

(grown in farm soil). Vegetative growths in all treatments were recorded till 90th day. The

tallest lady’s finger plant was observed in 100 gm/plant vermicompost of Notoscolex palniensis

(T1D) 60.41 ± 0.064 cm with good foliage and branching numbers 6.120 ± 0.113 (Table 1&

Fig.2). Gutierrez et al. (2007) reported that addition of vermicompost increased plant heights

and number of leaves in the yield of tomato (Lycopersicum esculentum) significantly which

confirms the results of the present study.


6

The number of branches shows a significant (6.120 ± 0.113) variation over that of control

(2.115 ± 0.163) in all the treated plants (Table 3). According to Forde and Lorenzo (2001)

root growth and branching is favored in nutrient-rich environment and in the presence of

hormones like auxins that enable the plant to optimize the exploitation of the available resources

which are in turn transformed into photo assimilates and transported again to the root

consequently influencing plant growth and morphology.

Yield

The weight and length of fruit and number of fruits / Bhendi plant is one of the most important

yield contributing traits in lady’s finger and was found maximum when treated with 100 gm

vegetable refuse (T1) vermicompost of Priodocheta pellucida 270.44 ± 0.134 kg fruit/plant

(T1C), Notoscolex palniensis 250.4 ± 0.163 (T

1D) followed by Lampito mauritii 150.6 ±

0.219 (T1A) and Octochetona pattoni 200.75 ± 0.176 (T

1B) vermicompost (Fig.1&3). Average

number of fruits/plant was Lemnoscolex scutarius 150.5 ± 0.141 (T1E) and Hoplochetella

stuarti 100.8 ± 0.134 (T1F) (Fig.1). Azarmi et al., (2008) studied on tomato (Lycopersicum

esculentum var. Super Beta) and the results of their study supported the findings of our study

that vermicompost has positive effect on growth, yield and elemental contents of plant as

compared to control. In the present study, number of fruits per plant was greatly influenced by

the treatment T1. Chand et al. (2008) experimented on tomato plants to find out the effect of

natural fertilizers on their yield and quality. They found that significantly highest yield was recorded

in the treatment receiving enriched vermicompost along with 3 sprays of liquid manure. Similar

trend was observed in development of fruit length and fruit girth. They also confirmed that

earthworms significantly improve plant growth.

Different doses of vermicompost produced different responses in A. esculentus and showed

maximum positive effect on growth parameters. The results corroborate the findings that different

doses of vermicompost caused different responses in growth parameters of L. esculentum

plant (Azarmi et al. 2008). Joshi and Vig (2010) reported significant increase in growth

parameters with application of vermicompost in L. esculentum. Vermicompost has influenced

the plant growth parameters like plant length, number of branches, number of flowers per

plant, number of fruits per plant, yield of fruit per plant, length of fruits were significantly


7

influenced in plots receiving different doses of different genus of vermicompost. According to

Gonzalez et al. (1996) and Tien et al. (2000) tree plantations may influence earthworm

abundance by altering the physico-chemical properties of soils viz. temperature, moisture

regime, pH, organic matter content and litter inputs. Overall plant growth with vermicompost

application has been reported in different studies (Arancon et al., 2006; Zaller 2007; Bachman

and Metzger 2008; Singh et al. 2008). Arancon et al. (2004, 2006); Bachman and Metzger

(2008) also reported growth and yield improvement in different crops with vermicompost

application. The results clearly indicated that the plants receiving vermicompost had produced

more fruits/branches, branches/plant, large sized fruits with higher total yield than those of

control.

Plot soil tests for physico chemical parameters (N, P, K, pH, EC and micronutrients) was

done before and after cultivation and the results were statistically correlated (Table 4). The

vermicompost prepared by all the three earthworm species showed a substantial difference in

total N content, which could be attributed directly to the species specific feeding preference of

individual earthworm species and indirectly to mutualistic relationship between ingested

microorganisms and intestinal mucus (Suthar and singh 2008). The worms during

vermicomposting converted the insoluable P into soluable forms with the help of P-solublizing

microorganisms through phosphates present in the gut, making it more available to plants

(Suthar and singh 2008, Padmavathiamma et al., 2008). Vermicomposting proved to be an

efficient process for recovering higher K from organic waste (Suthar and singh 2008). The

present findings corroborated to those of Delgado et al., 1995, who demonstrated that higher

K concentration in the end product prepared from sewage sludge. pH was neutral being

around 7 and and increased gradually from substrate to compost to vermicompost

(Nagavallemma et al., 2006,). Fares et al., 2005, found the increased pH at the end of the

composting process, which was attributed to progressive utilization of organic acids and increase

in mineral constituents of waste. The increased EC during the period of the composting and

vermicomposting processes is in consistence with that of earlier workers (Jadia and Fulekar

2008), which was probably due to the degradation of organic matter releasing minerals such

as exchangeable Ca, Mg, K and P in the available forms, that is, in the form of cations in the

vermicompost and compost (Tognetti et al., 2005).


8

The values of plant height, number of fruits, yield, and length of fruit with the soil parameters

were statistically analysed using regression ‘B’ value is positive for all parameters, because of

the application of vermicompost, height, fruit length and yield of fruit also (positive result)

increased. The low earthworm diversity observed is consistent with other studies on invertebrate

ecology in urban areas (Paul and Meyer, 2001). According to Paoletti (1999) and Curry et al.

(2002), earthworm populations in cultivated land are generally lower than those found in

undisturbed habitats. Agricultural activities such as ploughing, several tillage operations, fertilizing

and application of chemical pesticides have dramatical effect on invertebrate animals. Any

management practices applied to soil are likely to have some (positive or negative) effects on

earthworm abundance and diversity. These effects are primarily the result of changes in soil

temperature, soil moisture and organic matter quantity or quality (Hendrix and Edwards, 2004).

The abundance of earthworms may increase due to some agricultural activities like liming,

organic fertilizing etc. (Kõlli, Lemetti, 1999). Lavelle and Spain (2001) admit that the regional

abundance of earthworms and the relative importance of the different ecological categories

are determined by large scale climatic factors (mainly temperature and rainfall) as well as by

their phylogenetic and bio geographical histories together with regional parameters such as

vegetation type and soil characteristics. According to Hole et al (2005) the evidence from

comparative studies under arable regimes indicates a general trend for higher earthworm

abundance under organic management.

Vermicompost contains more nutrients in plant available forms such as phosphates, exchangeable

calcium, soluble potassium and other macronutrients with huge quantity of beneficial

microorganisms, vitamins and hormones which have influence on the growth and yield of plants

(Theunissen et al. 2010). Kumari and Ushakumari (2002) reported that enriched vermicompost

was a superior treatment for enhancing uptake of N, P, K, Ca and Mg by cowpea.

Vermicompost have been recognized as having considerable potentials as soil amendments.

Vermicomposts are products of depredated organic matter through interactions of earthworms

and microorganisms. The process accelerates the rete of decomposition of the organic matter,

alters the physical and chemical properties of the material and lowers the C: N ratio leading to

a rapid humification process in which the unstable organic matter is fully oxidised and stabilized

(Albanell et al., 1998). The application of organic manures brings about structural improvement


9

regeneration of soil structures and increasing the aeration within. It may cause the roots to

extend into a large volume of soil in addition to the increase of water retention in the soil profile

(Agarwal et al., 1995). The analysis of soil applied with fertilizer showed that it has all kinds of

nutrients needed for the better growth of the crop. The soil properties such as pH, EC, available

nitrogen, phosphorus, potassium, iron zinc, copper and manganese were found to vary in the

soils treated with vermicompost application (Chidambaram et al., 2013). This is also in

concordance with the present investigation that the soil nature differs entirely on the application

of vermicompost of various treatments. The results of this study indicate that incorporation of

vermicompost of plant origin into a traditional base medium of farm soil and sand enhanced

growth of bhendi plant through, at least in part, improved mineral nutrition.

References

1. Agarwal G.C, Sidhu A.S, Sekhon N.K, Sandhu K.S and Sur H.S.(1995) Soil Till. Res.,

Vol.36, pp.129-139.

2. Albanell E. Plaixats J and Cabrero T (1998) Biol.Fertil.Soils, Vol.6, pp. 266-269.

3. Arancon, N. Q., Edwards, C. A., Bierman, P., Welch, C., & Metzer, J. D. (2004).

Influence of Vermicomposts on field strawberries: effect on growth and yields. Bioresource

Technology, 93, 145-153.

4. Arancon, N. Q., Edwards, C. A., & Bierman, P. (2006). Influences of vermicomposts

on field strawberries: effects on soil microbial and chemical properties. Bioresource

Technology, 97, 831-840.

5. Azarmi, R., Giglou, M. T., & Taleshmikail, D. (2008). Influence of vermicompost on soil

chemical and physical properties in tomato (Lycopersicum esculentum) field. African

Journal of Biotechnology, 7(14), 2397-2401.

6. Bachman, G. R., and Metzger, J. D. (2008). Growth of bedding plants in commercial

potting substrate amended with vermicompost. Bioresource Technology, 99, 3155-3161.

7. Chand, S., Ali, T., Wani, J.A. and Masih, M.R., (2008). ‘Integrated Nutrient Resource

Management (INRM) for Sustainable Agriculture’, Proceedings of International


10

Symposium on Natural Resource Management in Agriculture. December19-20, 2008,

Jaipur, Rajasthan, India, pp. 51.

8. Chaudhuri, P.S., T.K.Pal, G.Battachrjee and S.K. Dey: (2003). Rubber leaf litters (Hevea

brasilensis, var RRIM 600) as vermiculture substrate for epigeic earthworms. Perionyx

excavates, Eudrilus eugeniae and Eisenia fetida. Pedobiologia, 47, 796-800.

9. Chaudhuri, P.S., T.K. Pal, S.Nath and S.K.Dey: (2012). Effects of five earthworm

species on some physico-chemical properties of soil. J.Environ.Biol., 33, 713-716.

10. Chidambaram AL.A, Mathivanan S, Kalaikandhan R and Sundramoorthy P. (2013)

Asian Journal of Plant Science and Research, Vol. 3(2), pp.15-22.

11. Curry, J. P., Byrne, D., Schmidt, O. (2002) Intensive cultivation can drastically reduce

earthworm populations in arable land. – Eur. J. Soil Biol. 38, p. 127-130.

12. Delgado M. Bigeriego, I. Walter and Calbo R. (1995) “Use of california red worm in

sewage sludge transformation”, Turrialba, Vol.45, pp.33-41.

13. Divya UK (2001). Relevance of vermiculture in sustainable agriculture. Agriculture

General –World July 2001, 9-11.

14. Dominguez J (2004). State of the art and new prospectives on vermicomposting research.

In: Edwards CA (Ed) Earthworm Ecology (2nd Edn), CRC Press LLC, Boca Raton, FI

USA, pp 401-424.

15. Dominguez J, Edwards CA (2004). Vermicomposting organic wastes: a review. In: Hanna

SHS, Mikhail WZA (Eds) Soil Zoology for Sustainable Development in the 21st Century,

Cairo, pp 369-395.

16. Edwards, L., J.R. Burney, G. Richter and A.H. MacRae, (2000). Evaluation of compost

and straw mulching on soil-loss characteristics in erosion plots of potatoes in Prince

Edward Island, Canada. Ecosystem and Environment, 81: 217-222.

17. Fares F., Albalkhi A., Dec.J, Bruns. M.A and Bollag.J.M (2005) “Physicochemical

characteristics of animal and municipal wastes decomposed in arid soils,” Journal of

Environmental quality, vol.34, No.4, pp.1392-1403.


11

18. Forde, B., and Lorenzo, H. (2001). The nutritional control of root development. Plant

Soil, 232, 51-68.

19. Gajalakshmi. S and Abbasi S.A (2004) Earthworms and vermicomposting, Indian J

Biotechnol, 3 486-494.

20. Gaudette, H.E., W.R. Flight, L. Toner and D.W. Folger, (1974). An inexpensive titration

method for the determination of organic carbon in recent sediments. J. Sediment Petrol.,

44: 249-253.

21. Giller, K.E, M.H.Beare, P. Lavelle, A.M.N. Izac and M.J. Swift: (1997). Agricultural

intensification, soil biodiversity and agroecosystem function. Applied soil Ecology, 6,

3-16.

22. Gonzalez, G., X.M. Zou and S. Borges: (1996). Earthworm abundance and species

composition in abandoned tropical crop lands: Comparisons of tree plantations and

secondary forests. Pedobioligia, 40, 385 -391.

23. Gutiérrez, M., Federico, A., Santiago-B., Jorge, M. M., Joaquin A., Carlos, N., Camerino,

A., Miguel, O. L., Maria, A., Rincón, R. and Dendooven, L. (2007) ‘Vermicompost as

a soil supplement to improve growth, yield and fruit quality of tomato (Lycopersicum

esculentum)’, Bioresource Technology, Vol.98, No .15 ,pp. 2781-2786.

24. Hendrix, P. F. and Edwards, C. A. (2004) Earthworms in Agro ecosystems: research

approaches. – Earthworm Ecology. Ed. C. A. Edwards, 2nd edition, CRC Press, Boca

Raton, London, New York, p. 287-295.

25. Hole, D.G., Perkins, A.J., Wilson, J.D., Alexander, I.H., Grice, P.V., Evans, A.D.(2005)

Does organic farming benefit biodiversity? Biological Conservation 122 113-130.

26. Ismail, S.A., (2005). The Earthworm Book. Other India Press, apusa, Goa, pp: 101.

27. Jackson, M.L., 1958. Soil Chemical Analysis, Prentice Hall Inc, Englewood Cliffs, New

Jersey, USA, pp: 498.


12

28. Jadia C.D and Fulekar M.H, (2008) “Vermicomposting of vegetable wastes: a biophysico

chemical process based on hydro-operating bioreacter,” African journal of Biotechnology,

vol.7, pp.3723-3730.

29. Joshi, R., & Vig, A. P. (2010). Effect of vermicompost on growth, yield and quality of

tomato (Lycopersicum esculentum L). African Journal of Basic and Applied Science 2,

(3-4), 117-123.

30. Kavdir, Y and R.Ilay: (2011). Earthworms and soil structure. In: Biology of earthworms

(Ed. : A.Karaca). Spiringer, London, p.39.

31. Kõlli, R., Lemetti, I. (1999) Eesti muldade lühiiseloomustus. I. Normaalsed

mineraalmullad. – Eesti Põllumajandusülikool, Tartu, 122 lk.

32. Kumari, M.S. and K. Ushakumari, (2002). Effect of vermicompost enriched with rock

phosphate on the yield and uptake of nutrients in cowpea (Vigna unguinculata L. WALP).

J. Trop. Agric.,40: 27-30.

33. Lavelle, P. and Spain, A.V. (2001). Soil Ecology. Dordrecht, Netherlands: Kluwer

Academic Publishers.

34. Lindsay, W.L. and W.A. Norvell, (1978). Development of DTPA soil test for zinc, iron,

manganese and copper. Soil Sci. Soc. Am. J., 42: 421-428.

35. Nagavallemma K.P, Wani S.P and Stephane L (2006) “Vermicomposting: recycling

wastes into valuable organic fertilizer, ” Journal of SAT Agricultural Research, Vol.2,

No.1, pp.1-17.

36. Neher, D.A.: (1999). Soil community composition and ecosystem process: comparing

agricultural ecosystems with natural Ecosystems. Agroforestry systems, 45, 159.

37. Ouedraogo, E., A. Mando and N.P. Zombré, (2001). Use of compost to improve soil

properties and crop productivity under low input agricultural system in West Africa.

Agric. Ecosys. Environ., 84: 259-266.

38. Paoletti, M. G. (1999) The role of earthworms for assessment of sustainability and as

bioindicators. – Agriculture. Ecosystems, Environment. 74, p. 37-155.


13

39. Padmavathiamma P.K, Li L.Y and Kumari U.R (2008) “An experimental study of

vermi-biowaste composting for agricultural soil improvement,” Bioresource technology,

Vol.99, No.6, pp.1672-1681.

40. Paul, M.J., Meyer, J.L. (2001). Streams in the urban landscape. Annu. Rev. Ecol.Syst.

32: 333-365.

41. Rhoades, J.D., N.A. Mantghi, P.J. Shause and W. Alves, (1989). Estimating soil salinity

from saturate soil-paste electrical conductivity. Soil Sci. Soc. Am. J., 53: 428-433.

42. Shipitalo, M. J. and Le Bayon, R.C. (2004). “Quantifying the effects of earthworms on

soil aggregation and porosity”. In Edwards, C.A. (ed.). Earthworm Ecology, second

edition. Boca raton, London, New York, Washington: CRC Press.

43. Singh, R., Sharma, R.R., Kumar, S., Gupta, R. K., & Patil, R. T. (2008). Vermicompost

substitution influences growth, physiological disorders, fruit yield and quality of strawberry

(Fragaria xananassa Duch). Bioresource Technology, 99(17), 8507-8511.

44. Suthar. S and Singh. S (2008) “Vermicomposting of domestic waste by using two epigeic

earthworms (Perionyx excavatus and Perionyx sansibaricus),” International Journal of

Environment Science and Technology, Vol.5, No.1, pp.99-106.

45. Theunissen, P. A., Ndakidemi, & Laubscher, C. P. (2010). Potential of vermicompost

produced from plant waste on the growth and nutrient status in vegetable production.

International Journal of Physical Science, 5(13), 1964-1973.

46. Tien, G, J.A. Olimah, G.O. Adeoye and B.T. Kang: (2000). Regeneration of earthworm

population in a degraded soil by natural and planted fallows under humid tropical condition.

Soil Sci. Sco. Am. J., 64, 222-228.

47. Tognetii C, Laos F, Mazzarino M.J and Hernandez M.T. (2005) “composting Vs

vermicomposting:a comparison of end product quality,” Compost Science and Utilization,

Vol.13, no.1,pp.6-13.

48. Zaller, J. G. (2007). Vermicompost as a substitute for peat in potting media: Effects on

germination, biomass allocation, yields and fruit quality of three tomato varieties. Science

of Horticulture, 112, 191-199.


14

Table : 1 Height of the Ladies finger plant (in cm) in

different dosages and different treatments

Three different dosages

Treatments

100 gm 200 gm 300 gm

Lampito mauritii (T1A) 58.42 ± 0.1344 50.45 ± 0.021 43.48 ± 0.035

Octochetona pattoni (T1B) 60.29 ± 0.071 51.46 ± 0.205 39.99 ± 0.141

Priodocheta pellucida (T1C) 53.59 ± 0.163 55.49 ± 6.371 43.77 ± 0.325

Notoscolex palniensis (T1D) 60.41 ± 0.064 45.95 ± 0.070 40.88 ± 0.169

Lemnoscolex scutarius (T1E) 50.33 ± 0.297 45.34 ± 0.148 30.23 ± 0.106

Hoplochetella stuarti (T1F) 40.60 ± 0.0566 37.66 ± 0.070 35.36 ± 6.561

Control (T1) 21.58 ± 0.085

Table : 2 Fruit length (in cm) of the Ladies finger plant in

different dosages and different treatments

Three different dosages

Treatments

100 gm 200 gm 300 gm

Lampito mauritii (T1A) 20.11 ± 0.148 18.94 ± 0.092 9.865 ± 0.148

Octochetona pattoni (T1B) 21.49 ± 0.077 19.26 ± 0.085 10.15 ± 0.078

Priodocheta pellucida (T1C) 20.99 ± 0.014 19.14 ± 0.155 11.495 ± 0.219

Notoscolex palniensis (T1D) 22.12 ± 0.162 20.63 ± 0.255 12.19 ± 0.233

Lemnoscolex scutarius (T1E) 19.77 ± 0.332 17.71 ± 0.297 7.740 ± 0.297

Hoplochetella stuarti (T1F) 13.37 ± 0.120 10.44 ± 0.297 6.93 ± 0.0989

Control (T1) 11.15 ± 0.120


15

Table : 3 No. of branches and No. of fruits of the

Ladies finger plant in in different dosages and different treatments

Treatments100 gm 200 gm 300 gm 100 gm 200 gm 300 gm

Lampito mauritii (T1A) 6.025 ±

0.035

5.485 ±

0.049

3.285 ±

0.078

12.40 ±

0.070

8.485 ±

0.233

4.950 ±

0.070

Octochetona pattoni (T1B) 4.865 ±

0.120

3.485 ±

0.049

2.955 ±

0.077

15.81 ±

0.198

10.96 ±

0.077

7.780 ±

0.311

Priodocheta pellucida (T1C) 5.965 ±

0.049

4.210 ±

0.297

2.570 ±

0.240

20.98 ±

0.035

12.95 ±

0.070

6.225 ±

0.318

Notoscolex palniensis (T1D) 6.120 ±

0.113

5.905 ±

0.049

3.7500

± 0.141

18.77 ±

0.332

10.22 ±

0.141

8.760 ±

0.226

Lemnoscolex scutarius (T1E) 4.700 ±

0.197

3.256 ±

0.276

2.185 ±

0.092

11.46 ±

0.106

7.800 ±

0.169

4.980 ±

0.028

Hoplochetella stuarti (T1F) 2.115 ±

0.163

1.147 ±

0.00014

0.541 ±

0.0064

8.425 ±

0.134

5.300 ±

0.085

2.155 ±

0.219

Control (T1) 2.115 ± 0.163 7.775 ± 0.289

No. of branches No. of fruits

Three different dosages Three different dosages

Table : 4. Occurrence of soil physico-chemical parameters

before and after vermicompost application with mean values

Nature of soil Physico-

chemical

parameters Before cultivation

control soil

After

vermicompost

applied soil

N 69.00 ± 2.65 76.00 ± 2.00

P 11.00 ± 5.00 25.33 ± 17.009

K 87.66 ± 13.65 129.0 ± 19.31

pH 6.99 ± 0.94 7.35 ± 0.22

EC 0.27 ± 0.16 0.69 ± 0.481

Fe 9.65 ± 2.52 41.0 ± 9.49

Mn 12.75 ± 3.33 17.91 ± 3.68

Zn 0.25 ± 0.012 0.553 ± 0.258

Cu 0.566 ± 0.241 2.93 ± 1.96


16

Figure : 1 yield of fruit (in gm) in Ladies finger plant in

different treatments and different dosages

Fig 2 : Measurement of fruit (Abelmoschus esculentus) length (in cm)


17

Fig 3 : A.esculantus plant fruit yield on different treatments Lampito mauritii (T1A),

Octochaetona pattoni (T1B) and Priodocheta pellucida (T

1C) in 3 different dosages


18

Fig 4: Normal probability – Plot of Regression Standardized Residual


19

Volume 5

No. 2

pp. 19-30

September 2014

Sciencia Acta Xaveriana

An International Science Journal

ISSN. 0976-1152

Maximum Independent Set Cover

Pebbling Number of an m-ary Tree

A. Lourdusamy1, C. Muthulakhmi @ Sasikala2 and T. Mathivanan3

1, 3 Department of Mathematics

St. Xavier’s College,

Palayamkottai - 627 002.

1 [email protected]

3 [email protected]

2 Assistant Professor in mathematics,

Sri Paramakalyani College,

Alwarkurichi, India.

Abstract : A pebbling move is defined by removing two pebbles from

some vertex and placing one pebble on an adjacent vertex. A graph is said

to be cover pebbled if every vertex has a pebble on it after a series of

pebbling moves. The maximum independent set cover pebbling number of

a graph G is the minimum number, , of pebbles required so that any

initial configuration of pebbles can be transformed by a sequence of

pebbling moves so that after the pebbling moves the set of vertices that

contains pebbles form a maximum independent set S of G. In this paper, we

determine the maximum independent set cover pebbling number of an

m-ary tree.

Key words: Graph pebbling, cover pebbling, maximum independent set

cover pebbling, m-ary tree.

20

1. Introduction

Given a graph G, distribute k pebbles on its vertices in some configuration, call it as

C. Assume that G is connected in all cases. A pebbling move is defined by removing

two pebbles from some vertex and placing one pebble on an adjacent vertex. [1] The

pebbling number is the minimum number of pebbles that are sufficient, so that

for any initial configuration of pebbles, it is possible to move a pebble to any

root vertex v in G. [2] The cover pebbling number is defined as the minimum

number of pebbles needed to place a pebble on every vertex of the graph using a

sequence of pebbling moves, regardless of the initial configuration. A set S of

vertices in a graph G is said to be an independent set (or an internally stable set) if

no two vertices in the set S are adjacent. An independent set S is maximum if G has

no independent set with .

We introduce the concept maximum independent set cover pebbling number in [4].

The maximum independent set cover pebbling number, , of a graph , to be the

minimum number of pebbles that are placed on such that after a sequence of

pebbling moves, the set of vertices with pebbles forms a maximum independent set

S of G, regardless of their initial configuration. In this paper, we determine the

maximum independent set cover pebbling number for an m-ary tree.

Notation: denotes the number of pebbles placed at the vertex . Also

denotes the number of pebbles on the graph .

2. Maximum independent set cover pebbling number of an m-ary tree

Definition 2.1. A complete -ary tree, denoted by , is a tree of height with

vertices at distances i from the root. Each vertex of has children except

Maximum Independent Set Cover Pebbling Number of an m-ary Tree

21

for the set of vertices that are at distance away from the root, none of which

have children. The root is denoted by .

Or Simply a complete -ary tree with height , denoted by , is an -ary tree

satisfying that has children for each vertex not in the th level.

Theorem 2.2. (i) ρ(M0) = 1 (obvious).

(ii) ρ(M1) = 4m-3 (m ≥ 3) and if m = 2 then ρ(M1) = 6. Since, for m ≥ 3, M1 ≡

K1,m[4] and for m = 2, M1 ≡ P3, the path of length two[5].

(iii) ρ(M2) = 16m2-12m+1.

Proof of (iii). Note that M2 contains m-M1’s as subtrees which are all connected to

the root R2 of M2. Let R11, R12, … , R1m be the root of the m-M1’s (say M11, M12, …

, M1m). In general, Mn contains m-Mn-1’s as subtrees which are all connected to the

root Rn of Mn. Let R(n-1)1, R(n-1)2, … , R(n-1)m be the root of the m-M(n-1)’s. Choose the

rightmost vertex of this subtree, label it by v. Put 16m2-12m pebbles on this vertex.

Then we cannot cover the maximum independent set of M2. Thus ρ(M2) ≥ 16m2-

12m+1.

Now consider the distribution of 16m2-12m+1 pebbles on the vertices of M2.

According to the distribution of these amounts of pebbles, we find the following

cases:

Case 1 : f(M1i) ≥ 4m-3, where 1 ≤ i ≤ m.

Clearly we are done if f(R2) ≥ 1. So assume that, f(R2) = 0. This implies that

2

1

1

( ) 16 12 1.m

i

i

f M m m=

= − +∑ Any one of the m2

paths (of length two) leading

from the root R2 to the bottom of M2 must contain at least four pebbles and hence

we are done, since any one the subtree contains at least


22

we are done, since any one the subtree contains at least 216 12 1

16 12 1m m

mm

− +≥ − +

pebbles.

Case 2: f(M1i) ≤ 4m-4, for all i (1 ≤ i ≤ m)

This implies that f(R2) ≥ 16m2-12m+1-m(4m-4)=12m

2-8m+1. We need 2m(4m-3)+1

pebbles at R2. But f(R2)- 2m(4m-3)- 1>0.

Case 3 : f(M1i) ≥ 4m-3 for some i (1 ≤ i ≤ m).

Let t ≥1 subtrees of M2 contains at least 4m-3 pebbles. Note that, for every subtree

(except one subtree that contains 4m-3 or more pebbles, we have 16m pebbles to

cover its maximum independent set.

Let f('

1 jM ) = aj where aj ≤ 4m-4. Thus, to cover the maximum independent set of

the subtree '

1 jM , we have another 16m-aj pebbles somewhere on the graph. So, we

can send 16

44 4

j jm a a

m−

≥ −

pebbles to the root R2 and then we move

28

ja

m − pebbles to the root '

1 jR of '

1 jM . Thus '

1 jM contains aj+2m-8

ja

=

2m+7

8ja . But these numbers of pebbles are enough to cover the maximum

independent set of '

1 jM , or the value of 2m+7

4 38

ja m≥ − , and hence we are

done. So using (m-t)(16m-aj)- 1

t

j

i

a=

∑ pebbles, we cover the maximum independent

set of the (m-t) subtrees that contains aj pebbles. So we have at least (t-1)16m+4m+1


23

pebbles on the t-subtrees plus R2 that are all contains 4m-3 or more pebbles. If f(R2)

≥ 1 then we are done. Otherwise we can always move a pebble to R2 using at most

four pebbles from the remaining pebbles on the t-subtrees.

(iv) ρ(M3) = 64m3-

48m

2+4m-15 (m ≥ 3).

Proof of (iv). Clearly, M3 contains m-M2’s as subtrees which are all connected to

the root R3 of M3. Consider the rightmost bottom vertex, say v, of M3 and put 64m3-

48m2+4m-16 pebbles on the vertex v. Then we cannot cover the maximum

independent set of M3. Thus ρ(M3) ≥ 64m3- 48m

2+4m-15.

Now consider the distribution of 64m3-

48m

2+4m-15 pebbles on the vertices of M3.

According to the distribution of these amounts of pebbles, we find the following

cases:

Case 1 : f(M2i) ≥ ρ(M2) where 1 ≤ i ≤ m.

Clearly we are done if f(R2) =0, or 2 or f(R2) ≥ 4. So assume that f(R2) = 1 or 3. This

implies that, 3 2

2

1

( ) 64m - 48m +4m-18 .m

i

i

f M=

≥∑ pebbles. So, any one of the

path (of length three) leading from the root R3 to the bottom row of M3 must contain

at least eight pebbles. Thus we move a pebble to R3 and hence we are done.

Case 2 : f(M2i) < ρ(M2) where 1 ≤ i ≤ m.

We need 2m ρ(M2)+5 pebbles on the root vertex R3 of M3. We have ρ(M3)-m

ρ(M2)+m pebbles on the root vertex R3. But, ρ(M3)-m ρ(M2)+m-(2m ρ(M2)+5) ≥ 0.

Since, ρ(M3) = 64m3-

48m

2+4m-15, ρ(M2) = 16m

2-12m+1 and m ≥ 3.

Case 3 : f(M2i) ≥ ρ(M2) for some i (1 ≤ i ≤ m).


24

Let t ≥ 1 subtrees contains ρ(M2) or more pebbles. Label those subtrees by M2i (1 ≤ i

≤ t) and label the other subtrees by '

2 jM (1 ≤ i ≤ m-t). Also, let f('

2 jM ) = aj where

aj < ρ(M2). Note that, we have usually (64m2+16)(m-1) pebbles each to cover the

maximum independent set of M2i’s and '

2 jM ’s, except one subtree M2k (1 ≤ k ≤ t)

that contains ρ(M2) or more pebbles.

Since aj < ρ(M2), we have another 64m2+16-aj pebbles that are in somewhere of the

graph M3 to cover the maximum independent set of '

2 jM . So we can send

2

j 264m +16-a

8 28 8

jam

≥ + −

pebbles to the root R3 and then we move

4m2+1- pebbles to the root

'

2 jR of '

2 jM . Thus, '

2 jM contains 2 15

4 116

jm a+ +

pebbles. But these number of pebbles are at least ρ(M2) or it is enough to cover the

maximum independent set of '

2 jM using the pebbles at

'

2 jR plus aj pebbles. Thus

the t-subtrees M2i plus R3 contains (64m2+16)(t-1)+ 16m

2-12m+1 or more pebbles.

We know that f(M2i) ≥ ρ(M2) where 1 ≤ i ≤ t. Let f(R3) =1 or 3 (Otherwise, we are

done). We can move a pebble to R3, using at most eight pebbles from the subtree

that contains 16m2-12m+9 pebbles or more. And hence we are done.

(v) .

Proof of (v): Consider the rightmost bottom vertex, say v, of M4, and put

pebbles. Then we cannot cover the maximum

independent set of M4. Thus, .


25

Now consider the distribution of pebbles on the

vertices of M4. According to the distribution of these amounts of pebbles, we find

the following cases:

Case 1: for all i (1 ≤ i ≤ m).

Clearly we are done if . So assume that . This implies that

. So any one of the m4

paths (of length four) leading from the root R4 to the bottom row of M4 contains at

least sixteen ‘extra’ pebbles. Thus we can move a pebble to R4 and hence we are

done.

Case 2: for all i (1 ≤ i ≤ m).

We need pebbles on the root vertex R4 of M4. We have

on the root vertex R4. Since,

,

and we get and hence we are done.

Case 3: for some i.

Similar to Case (iii) of previous theorems; using the hints, from that

pebbles, we can send to the root R4 of M4.

Theorem 2.3: For a complete m-ary tree Mn (n ≥ 3), the maximum independent set

cover pebbling number is given by,


26

where , and

.

Proof. Consider the rightmost vertex of Mn, say v, and put pebbles on the

vertex v. Then we cannot cover a maximum independent set of Mn. Thus the lower

bound follows.

We prove the upper bound of by induction on n. For n=3 and n=4, this

theorem is true by previous theorem (iv) and (v). So assume the result is true for the

complete m-ary tree Mn-1 (n ≥ 5).

Consider the distribution of pebbles on the vertices of Mn. According to the

distribution of these amounts of pebbles, we find the following cases:

Case (1): for all i (1 ≤ i ≤ m).

We need, pebbles on the root Rn, to cover the maximum

independent set of Mn. We have to prove that

. It is enough to prove that,

(for m ≥ 3).

------ (1)

From the 1st term, by considering k=0 we get,

------ (2)


27

------ (3)

------ (4)

and ------- (5)

Equation (2) through (5) show that (1) holds if,


28

which holds for and . Also, for and

.

Case (2): for all i (1 ≤ i ≤ m).

Subcase 2.1: n is odd.

If then clearly we are done. So assume that

. Then, or more pebbles on the . We know

that, and . We have,

extra pebbles on the vertices of . Thus at least one

subtree contains extra pebbles, so at least

one of the paths leading to the root from the bottom of the subtree has at

least 2n pebbles and hence we are done.

Subcase 2.2: n is even.

If then we are done. So assume that . Like, Subcase 2.1, at

least one of the paths has 2n or more pebbles and hence we are done.

Case (3): for some i.


29

Let subtrees contain or more pebbles. Label those subtrees by

and label the other subtrees by . Also let

where . Clearly we can supply at

least one pebble to the root of for every 2n extra pebbles on

. Also, having one additional pebble in is

equivalent to have at least one pebble on the root vertex of .

Note that, we have usually used P pebbles each to cover the maximum independent

set of and , except one subtree, say

, that contains or more pebbles. Since , we have

extra pebbles, that are in somewhere of the graph , to cover the maximum

independent set of . So we can send pebbles to

the root vertex of . Thus contains

pebbles. But these amounts of pebbles are at least

or it is enough to cover the maximum independent set of , using the

pebbles at plus pebbles. Thus the t-subtrees plus

contains or more pebbles. We know that

where .

Subcase 3.1: n is odd.

Let (otherwise we are done easily). Then we can move a pebble to


30

, using at most pebbles from the subtree that contains at least

pebbles and hence we are done [since ].

Subcase 3.2 : n is even.

Let (otherwise we are done). Like the Subcase 3.1, we can move a pebble

to , using at most pebbles (from the subtree that contains

pebbles or more).

References :

[1] F.R.K. Chung, Pebbling in hypercubes, SIAM J. Disc. Math 2(1989), 467-472.

[2] B.Crull, T.Cundiff, P.Feltman, G.H. Hurlbert, L.Pudwell, Z.Szaniszlo, Z.Tuza, The

cover pebbling number of Graphs, (2004).

[3] G.Hurlbert, A survey of Graph Pebbling, Congressus Numerantium 139 (1999)

41-64.

[4] A. Lourdusamy, C. Muthulakshmi @ Sasikala and T. Mathivanan, Maximum

independent set cover pebbling number of a Binary Tree, Sciencia Acta Xaveriana,

Vol. 3(2) (2012) , 9-20.

[5] A. Lourdusamy, C. Muthulakshmi @ Sasikala, Maximum independent set cover

pebbling number of a Star, International Journal of Mathematical Archive- 3(2),

2012, 616-618.

[6] A. Lourdusamy, C. Muthulakshmi @ Sasikala and T. Mathivanan, Maximum

independent set cover pebbling number of complete graphs and paths, submitted for

publication.


31

Volume 5

No. 2

pp. 31-38

September 2014

Sciencia Acta Xaveriana

An International Science Journal

ISSN. 0976-1152

Generalized t-Pebbling Numbers of

Wheel and Complete r-partite graph

A. Lourdusamy

Department of Mathematics

St. Xavier’s College (Autonomous)

Palayamkottai - 627 002, India

[email protected]

C. Muthulakshmi@Sasikala

Department of Mathematics

Sri Paramakalyani College

Alwarkurichi - 627 412, India

Abstract : The generalized t-pebbling number of a graph G, fglt

(G), is the least

positive integer n such that however n pebbles are placed on the vertices of G, we

can move t-pebbles to any vertex by a sequence of moves, each move taking

p pebbles off one vertex and placing one on an adjacent vertex. In this paper, we

determine the generalized t-pebbling number of wheel Wn and complete r-partite

graph.

Key Words : Graph, wheel and complete r-partitle graph.

1 Introduction

Let G be a simple connected graph. The pebbling number of G is the smallest

number f(G) such that however these f(G) pebbles are placed on the vertices of G,

32

we can move a pebble to any vertex by a sequence of moves, each move taking two

pebbles off one vertex and placing one on an adjacent vertex [2]. Suppose n pebbles

are distributed on to the vertices of a graph G, a generalized p pebbling step [u,v]

consists of removing p pebbles from a vertex u, and then placing one pebble on an

adjacent vertex v, for any p ≥ 2. Is it possible to move a pebble to a root vertex r, if

we can repeatedly apply generalized p pebbling steps? It is answered in the

affirmative by Chung in [1]. The generalized pebbling number of a vertex v in a

graph G is the smallest number fgl(v,G) with the property that from every placement

of fgl(v,G) pebbles on G, it is possible to move a pebble to v by a sequence of

pebbling move consists of removing p pebbles from a vertex and placing one pebble

on an adjacent vertex. The generalized pebbling number of the graph G, denoted by

fgl(G), is the maximum fgl(G) over all vertices v in G.

Again the generalized t-pebbling number of a vertex v in a graph G is the smallest

number fglt(v,G) with the property that from every placement of fglt(v,G} pebbles on

G, it is possible to move t pebbles to v by a sequence of pebbling moves where a

pebbling move consists of the removal of p pebbles from a vertex and the placement

of one of these pebbles on an adjacent vertex. The generalized t-pebbling number

of the graph G, denoted by fglt(G) is the maximum fglt(v,G) over all vertices v of G.

Throughout this paper G denotes a simple connected graph with vertex set V(G) and

edge set E(G).

x denote the largest integer less than or equal to x and x denote the smallest

integer greater than or equal to x.

2 Known Results

We find the following results with regard to the generalized pebbling numbers of

graph in [2, 6] and their generalized t-pebbling numbers in [3].

Generalized t-Pebbling Numbers of Wheel and Complete r-partite graph

33

Theorem 2.1. For a complete graph Kn, fgl(Kn) = (p-1)n-(p-2) where p ≥ 2.

Theorem 2.2. For a path of length n, fgl(Pn) = pn where p ≥ 2.

Theorem 2.3. For a star K1,n, fgl(K1,n) = (p-1)n+(p2-2p+2) if n > 1 and p ≥ 2.

Theorem 2.4. The generalized t-pebbling number for a path of length n is

fglt(Pn)=tpn.

Theorem 2.5. The generalized t-pebbling number of a complete graph on n vertices

where n ≥ 3, p ≥ 2 is fglt(Kn) = pt+(p-1)(n-2).

Theorem 2.6. The generalized t-pebbling number for a star K1,n where n > 1 is

fglt(K1,n)=p2t+(p-1)(n-2) where p ≥ 2.

Theorem 2.7. For n ≥ 4, the generalized pebbling number of the wheel graph Wn is

fgl(Wn) = (p-1)+(p2-2p+1) where p ≥ 2.

Theorem 2.8. The generalized pebbling number of the fan graph Fn is fgl(Fn) = (p-

1)n+(p2-2p+1).

Theorem 2.9. For G = 1 2 rs ,s , ... ,s

K the generalized pebbling number is given by

fgl(G) =

2

1 1

1

p ( 1)( 2).

( 1)( 2)

p s if p n s

p p n if p n s

+ − − ≥ −

+ − − < −

We will now proceed to compute the genearlized t-pebbling numbers of wheel Wn

and complete r-partite graph.

3 Computation of Genearlized t-pebbling number

Definition 3.1. We define the wheel graph denoted by Wn to be the graph with

V(Wn)= {h,v1,v2, … ,vn} where h is called the hub of Wn and E(Wn)=E(Cn) ∪ {hv1,

hv2, … , hvn} where Cn denotes the cycle graph on n vertices.

A. Lourdusamy and C. Muthulakshmi@Sasikala

34

Theorem 3.2. Let K1 = {h}. Let Cn = {v1,v2, … ,vn} be a cycle of length n. Then the

generalized t-pebbling number of the wheel graph Wn is fglt(Wn) = p2(t-1)+(p-

1)n+(p2-2p+1).

Proof : By Theorem 2.5, fglt(h,Wn) = pt+(p-1)(n-1). Let us now find the generalized

t-pebbling number of v1. Assume that v1 has zero pebbles. Let us place (p2t-1)

pebbles at 2

nv

, (p-2) pebbles at vn and (p-1) pebbles at each of wn\{v1, 2

nv

, vn}.

Then t pebbles cannot be moved to v1.

So fglt(v1,Wn) ≥ p2(t-1)+(p-1)n+(p

2-2p+1).

Let us use induction on t to prove the fglt(v1,Wn) ≤ p2(t-1)+(p-1)n+(p

2-2p+1).

For t=1, the result is true by Theorem 2.7.

By distributing p2(m-2)+(p-1)n+(p

2-2p+1) pebbles on Wn \{v1}, then we can move

(m-1) pebbles to the target vertex v1.

That is, fgl(m-1)(Wn)=p2(m-2)+(p-1)n+(p

2-2p+1). Suppose p

2(m-1)+(p-1)n+(p

2-2p+1)

pebbles are distributed on to the vertices of Wn \ {v1}. Let the target vertex be v1 of

Cn.

0.

1

If there is a vertex in Cn with at least p2 pebbles, then a pebble can be moved to v1.

Using only p2 pebbles through h. The remaining p

2(m-2)+(p-1)n+(p

2-2p+1) pebbles

are sufficient to put (m-1) additional pebbles on v1 by using induction. Otherwise

any one of the vertices of Wn \ {v1} say

2

nv

receive at least p pebbles and each of

the vertices Wn\ {v1, 2

nv

} receive p-1 pebbles then from 2

nv

using a sequence of


35

pebbling moves, 2

nv

, 1

2

nv

−

, … ,v1 we can move a pebble to v1. Remaining p2+(p-1)

(n-2

n

+2)+(p2-3p+1) > 0. So by induction, (m-1) pebbles can be moved to v1.

Hence in all cases fglm(v1,Wn) ≤ p2(m-1)+(p-1)n+(p

2-2p+1). Therefore fglt(Wn)=p

2(m-

1)+(p-1)n+(p2-2p+1).

Definition 3.3. A graph G = (V,E) is called an r-partite graph if V can be partitioned

into r non-empty subsets V1,V2, … ,Vr such that no edge of G joins vertices in the

same set. The sets V1,V2, … ,Vr are called partite sets or vertex classes of G. If G is

an r-partite graph having partite sets V1,V2, … ,Vr such that every vertex of Vi is

joined to every vertex of Vj where 1 ≤ i, j ≤ r and i ≠ j, then G is called a complete r-

partite graph. If |Vi|=si for i=1,2, … , r then we denote G by 1 2 rs ,s , ... ,s

K .

Notation 3.4. For s1 ≥ s2 ≥ … ≥ sr, s1 > 1 and if r = 2, s2 > 1, let 1 2 rs ,s , ... ,sK be the

complete r-partitle graph with s1, s2, … , sr vertices in vertex classes C1, C2, … , Cr

respectively. Let n = 1

r

i

i

s=

∑ .

Theorem 3.5. For G = 1 2 rs ,s , ... ,sK the generalized t-pebbling number for a complete r-

partite graph G is given by

fglt(G) = 1

2

1 1

( 1)( 2).

p ( 1)( 2)

pt p n if pt n s

t p s if pt n s

+ − − < −

+ − − ≥ −

Proof :

Case i: Assume pt < n - s1.


36

Let us place pt+(p-1)(n-2)-1 pebbles on the vertices of G-{v} as follows. Let us

choose (t-1) vertices and we place p+(p-1) pebbles on each of the (t-1) vertices and

we place (p-1) pebbles each on the remaining vertices clearly t pebbles cannot be

moved to v.

Hence fglt(v,G) > (t-1)[(p+(p-1)]+(p-1)(n-t)

= pt+(p-1)(n-2)-1

≥ pt+(p-1)(n-2).

Next we will use induction to show that pt+(p-1)(n-2) pebbles are sufficient to move

t pebbles to any desired vertex. For t=1 results is true by Theorem 2.9. Suppose t >

s1, and pt+(p-1)(n-2) pebbles are placed on the vertices of G. Let the target vertex be

v of Ck for some k=1, 2, … , n. If there is a vertex w of Cj (j ≠ k) with at least p

pebbles then a pebble can be placed on v.

The remaining p(t-1)+(p-1)(n-2) pebbles are sufficient to put (t-1) additional pebbles

on v by induction. If not then every vertex of G\Ck wil have at most (p-1) pebbles on

it. Suppose among these n-sk vertices, q is the number of vertices with at least one

pebble. Therefore there will be pt+(p-1)(n-2)-q pebbles on the vertices of Ck. We

consider the following cases.

Subcase I : q ≥ t.

We use pebbling move from sk-1 vertices of Ck\{v} to put the remaining at most (p-

1) pebbles on each of the t of the q occupied vertices of v(G)-Ck. Using (p-1)t

pebbles we can pebble t vertices with (p-1) pebbles. Then remaining (p-1)(n-2)-(q-t)

pebbles are in Ck\{v}. From the t vertices with p pebbles we can move t pebbles to v.


37

Subcase ii : q < t.

As in subcase (i) first we will put (p-1) more pebbles on each of these q vertices by

maiing (p-1)q moves from the vertices of Ck\{v} in order to put q pebbles on v. Then

we have to place t-q additional pebbles on v. So we use p2(t-q)+(p-1)pq=p

2t-pq

pebbles among pt+(p-1)(n-2)-q pebbles in the vertices of Ck\{v}. Hence in all the

cases fglt(v,G) ≤ pt+(p-1)(n-2).

Case ii: Assume pt ≥ n - s1.

Let the vertices of C1 be v1, v2, … , vn and let 1s

v be the target vertex. Let us place

p2t+(p-1)(s_1-2) pebbles on the vertices of C1 as follows. Let us place p

2t-1 pebbles

on v1 and place (p-1) pebbles each on (s1-2) vertices of C1 other than v1 and 1s

v . In

this case t-pebbles cannot be moved to 1s

v . Hence fglt(G) ≥ p2t+(p-1)(s1-2).

Next we will use induction on t to prove that p2t+(p-1)(s1-2) pebbles are sufficient to

put t pebbles on any desired vertex clearly the claim is true for pt=n-s1.

Since by case(i) fglt(G) = pt+(p-1)(n-2)

= pt+(p-1)(pt+s1-2)

= p2t+(p-1)(s1-2).

Suppose p(m-1) > n-s1 and fgl(m-1)(G) = p2t(m-1)+(p-1)(s1-2) = p

2m+(p-1)s1-

(p2+2p+2).

We prove the result is true for m where pm > n-s1. Suppoe p2m+(p-1)(s1-2) pebbles

are distributed on the vertices of G. Let the target vertex be v of Ck. If there is a

vertex in some Cj (j ≠ k) with at least p pebbles, then a pebble can be placed on v

2


38

using only p pebbles. The remaining p2m+(p-1)s1-3p+2 pebbles are sufficient to put

(m-1) additional pebbles on v, since p2+2p-2-3p+2 > 0. If not then every vertex of

G\Ck will contain either zero or at least one pebble on it. If there is a vertex say w in

some Cj (j ≠ k) with at least one pebble on it, we use (p-1)p pebbles from the vertices

of Ck to put (p-1) pebbles on w and hence a pebble can be placed on v. Since p2+2p-

2-(p-1)(p+3) > 0, then remaining fgl(m-1)(G) pebbles would suffice to put (m-1)

additional pebbles on v. Otherwise, every vertex of G\Ck will have zero pebbles,

using p2 pebbles we can place a pebble on v in this case the remaining p

2(m-1)+(p-

1)(s1-2) pebbles would suffice to put (m-1) additional pebbles on v. Thus fglm(v,G) ≤

p2m+(p-1)(s1-2). Therefore by induction fglt(v,G) ≤ p

2t+(p-1)(s1-2) for all pt < n-s1.

Thus fglt(G) < p2t+(p-1)(s1-2) for all pt ≥ n-s1 and so the proof is over.

References :

[1] F.R.K.Chung, Pebbling in Hypercubes, SIAM J. Discrete Maths., Vol 2(4)(1989)

pp 467-472.

[2] G. Hurlbert, Recent Progress in graph pebbling, Graph Theory notes of New York

XLIX (2005), 25-34.

[3] A. Lourdusamy and C. Muthulakshmi@ Sasikala, Generalized Pebbling Number,

International Mathematical Forum, 5, 2010, No.27, pp.1331-1337.

[4] A. Lourdusamy and C. Muthulakshmi@ Sasikala, Generalized t-pebbling Number of

a Graph, Journal of Discrete Mathematical Sciences & Cryptography, Vol. 12 (2009),

No. 1, pp. 109-120.

[5] A. Lourdusamy and C. Muthulakshmi@ Sasikala, Generalized pebbling Numbers of

some Graphs, Sciencia Acta Xaveriana,Vol3, No.1, (20012), pp107-114.

[6] A. Lourdusamy and A. Punitha Tharani, On t-pebbling graphs, Utilitas Mathematica,Vol.

87,( 2012), pp.331-342.


39

Volume 5

No. 2

pp. 39-38

September 2014

Sciencia Acta XaverianaAn International Science JournalISSN. 0976-1152

Super Vertex Mean Graphs

1. Introduction

A vertex labeling1 of a graph G is an assignment ƒ of labels to the vertices of G that

induces a label for each edge uv depending on the vertex labels. An edge labeling of a

graph G is assignment ƒ of labels to the edges of G that induces a label for each

vertex v depending on the edge labels. Let G = (V, E) be a simple graph with p

vertices and q edges. A mean labeling ƒ is an injection from V to the set (0,1,2,…, q)

that induces for each edge uv the label ( ) ( )

2

u v

such that the set of edge labels is

(1, 2,…, q). Mean labeling was introduced by Somasundaram and ponraj [8]. A graph

that accepts a mean labeling is known as mean graph. A super mean labeling ƒ is an

injection from V to the set {1,2,…, p+q} that induces for each edge uv the label

( ) ( )

2

u v

such that the set of all vertex labels and the induced edge labels is

{1,2,…,p+q}. In this paper we study super vertex behavior of certain classes of graph.

Super vertex mean labeling was introduced by R.Ponraj et al.[7]. A graph that

A.Lourdusamy1, M.Seenivasan2, Sherry George3 and R.revathy4

Abstract : An edge labeling of a graph G is an assignment ƒ of labels

to the edges of G that induces a label for each vertex depending on

the edges labels. A super vertex mean labeling ƒ is an injection from E

to the set {1,2,3,...,p+q} that induces for each vertex the label

Round such that the set of all edge labels and the induced

vertex labels is {1,2,3,...,p+q}. In this paper we study super vertex

behavior of certain classes of graphs.

40

(1,2,…,p+q}. In this paper we study super vertex behavior of certain classes of graph.

Super vertex mean labeling was introduced by R.Ponraj et al.[7]. A graph that

accepts a super mean labeling is known as super mean graph. Some results on mean

labeling and super mean labeling are given in [4, 5, 6, 7, 8, 9]. For a summary on

various graph labeling see the Dynamic survey of graph labeling by Gallian [2].

Lourdusamy and Seenivasan [3] introduced vertex mean labeling as an edge

analogue of mean labeling as follows: A vertex mean labeling of a (p,q) graph G(V,E)

is defined as an injection f :E {0,1,... , q*}, q* = max(p,q) such that the injection f :

V N defined by the rule v(V) = Round

( )

( )

ev e

d v

satisfies the property that

v(V) = {

v (u): u } = {1,2,..., p}, where v denotes the set of edges in G that

are incident at v and N denotes the set of all natural numbers. A graph that has a

vertex mean labeling is called a vertex mean graph or V-mean graph. For all

terminology and notations in graph theory, we refer the reader to the text book by

D.B.West [10]. All graphs considered in this paper are finite and simple. Motivated

by the concept of super mean labeling, we introduce super vertex mean labeling of

graphs as follows

Definition 1.1. A Super vertex mean labeling f of a graph G(V,E) is an injection from E

to the set {1,2,3,...,p+q} that induces for each vertex v the label v(v) = Round

( )

( )

EV e

d v

such that the set of all edge labels and the induced vertex labels is

{1,2,3,... , p+q}.

Henceforth we call super vertex mean as SVM. To initiate the investigation we obtain

certain classes of graphs which are SVM graphs. It is obvious that no tree is an SVM

graph. We also observe that C4 is not an SVM graph.


41

1. Some SVM Graphs

Theorem 2.1. The cycle Cn is an SVM graph if and only if n ? 4.

Proof: Let {e1, e2, ..., en} be the edge set of Cn such that ei =vivi+1, ,

en = vnv1. Checking each of the possibilities reveals that the cycle C4 is not an

SVM. So we assume that n 4.

Case 1: n 1 (mod 2). Let n = 2r+1. The edges of Cn are labeled as follows:

f (ei) =

It is easy to observe that f is injective. The induced vertex labels are given as

follows

v(vi) =

It is clear that f (E) U v(V) = {2i – 1: 1 } U {

= {1,2,3, ..., 2n}.

Case 2: n let n = 2r. The edges of Cn are labeled as follows:


42

Hence the theorem.

Super vertex-mean labeling of C9 and c

10 are shown in Figure 1.

FIGURE 1. Super vertex-mean labeling of C9 and C10

7

9

12

14 5

16

18

3

1

10

2 17

15

13

11

4

8

6

16 20

17 12

7 13

6 3

1 6

5

4

8 2

10

14 19

18

15

11

Theorem 2.2

Prof. Let V(PnxP

2) = {u

1, u

2, ........, u

n} U {V

1.....v

2, ......vn) and E (P

n x P

2) = {U

iU

i+1 :

1 < i < n-1} U {ViV

i+1 : < i < n-1} U {u

iv

i : 1 < I < n}. We note that the order of L

n is 2n

and the size is 3n-2

v1 v2 v3 v4 v5

u5 u4 u3 u2 u1

21 1

2 6

22 4 8 13

11

18

16 20

5 10 14

FIGURE 2. Super vertex-mean labeling of L5

and L8

3 9 15 19 25 29 34

38 32 27 22 17 12 7

u8 u7 u6 u5 u4 u3 u2 u1

1

v8 v7 v6 v5 v4 v3 v2 v1

5 10 14 20 24 30 36

4 8 13 18 23 28 33 37

35 31 26 21 16 11 6 2

The edges of Ln are labeled as follows :


43

f (uiui+1) =

f (vivi+1) =

f (uivi) =

it is easily observed that f is injective. The induced vertex labels are as follows:

v(ui) =

v(vi) =

It is easy to verify, in both cases, that the set of all edge labels and the induced

vertex labels is {1,2,....,5n-2}.

Hence the theorem.

SVM labeling of L5 and L8 are shown in Figure 2.

Theorem 2.3. A triangular snake with n blocks is SVM.

Proof. Let Gn be a triangular snake with n blocks on p vertices and q edges. Then p

= 2n + 1 and q = 3n.

Let V(Gn) = {ui : U vi : and V(Gn) = {uiui+1, uivi, ui+1vi :

The order of Gn is 2n+1 and size is 3n.

The edges of Gn are labeled as follows :

f (uiui+1) =


44

f (uivi) =

f (viui+1) =

Then, the induced vertex labels are as follows:

v(ui)

v(vi)

It can be easily verified that f is injective and the set of edge labels and induced

vertex labels is {1,2,...., 5n+1}.

Hence the theorem.

Super vertex mean labeling of triangular snakes are shown in Figure 3.

19

21

17 15

12

13 9

10

7 5

1

3

2

4

6

8

11

14

16

18

20

23

19

26 19

20

17 15

12

13 9

10

7 5

1

3

2

4

6

8

11

14

16

18

21 24

25

FIGURE 3. Super vertex-mean labeling of triangular snakes Figure 3 : Super vertex-mean labeling of triangular snakes

Theorem 2.4. Quardrilateral snakes are SVM.


45

Proof. Let Gn be a quadrilateral snake with V(Gn) = ui : U ui, wi :

and E(Gn) = uiui+1, uivi, ui+1wi, viwi : . Then p = 3n + 1 and q =

4n. Define f: E(Gn) {1,2,3,..., 7n+1} as follows:

i i+1) =

i i) =

i i) =

i i+1) =

Then the induced vertex labels are as follows :

vi) =

vi) =

vi) =

It can be easily verified that is injective and the set of edge labels and induced

vertex labels I {1,2,3,..., 7n+1}.

Hence the theorem.

A Super vertex-mean labeling of a Quadrilateral snake is shown in Figure 4.


46

27

25

22

29

20

18

13

11

8

15

14

6

3

1

7

2

4

5

9

10 12

16

17 19

23

24 26

28

FIGURE 4. A Super vertex-mean labeling of a Quadrilateral snake

References :

[1] B.D.Acharya and K.A.Germina. Vertex-graceful Graphs. Journal of Discrete

Mathematical Science & Chryptography, Vol.13(2010), No.5, pp.453-463.

[2] J.A.Gallian. A Dynamic Survey of Graph Labeling. The electronic journal of

combinatorics, 18, 2011.

[3] A.Lourdusamy and M.Seenivasan. Vertex-mean Graphs. International journal of

Mathematical Combinatorics, 3:114-120, 2011.

[4] A.Lourdusamy and M.Seenivasan. Mean Labelings of Cyclic Snakes. AKCE

International Journal of Graphs and Combinatorics, 8(2): 105-113, 2011.

[5] R.Ponraj, Studies in Labelings of Graphs. Ph.D.thesis, Manonmaniam Sundaranar

University, India (2004).

[6] R.Ponraj and D.Ramya. Super Mean Labeling of Graphs, Preprint.

[7] D.Ramya, R.Ponraj, and P.Jeyanthi. Super Mean Labeling of Graphs, Ars Combin.,

to appear.

[8] S.Somasundaram and R.Ponraj. Super Mean Labeling of Graphs, Natl Acad., Sci.Let.,

26(2003) 210-213.

[9] R.Vasuki and A.Nagrajan. Some Results on Super Mean Labeling of Graphs.

International Journal of Mathematical Combinatorics, Vol.3 (2009) 82-96.

[10] D.B.West. Introduction to Graph Theory. Prentice-Hall of India, Private Limited, New

Delhi, 1996.


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