some families of directed strongly regular graphs obtained ... - olmez.pdf · graphs obtained from...

Directed strongly regular graphs(DSRG)Combinatorial Designs

Construction of Directed Strongly Regular GraphsReferences

Some Families of Directed Strongly RegularGraphs Obtained from Certain Finite Incidence

Structures

Oktay Olmez

Department of MathematicsIowa State University

24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

May 12, 2011

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Overview

1 Directed Strongly Regular Graphs

2 Combinatorial Designs

3 Directed Strongly Regular Graphs Obtained from Affine Planes

4 Directed Strongly Regular Graphs Obtained from TacticalConfigurations

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Overview





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Overview





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Overview





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Directed strongly regular graphs(DSRG)

Definition

A loopless directed graph D with v vertices is called directedstrongly regular graph with parameters (v , k , t, λ, µ) if and only ifD satisfies the following conditions:

- Every vertex has in-degree and out-degree k .

- Every vertex x has t out-neighbors, all of which are alsoin-neighbors of x .

- The number of directed paths of length two from a vertex xto another vertex y is λ if there is an edge from x to y , and isµ if there is no edge from x to y .

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Definition





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Definition





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Definition





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DSRG-(8,3,2,1,1)

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Tactical Configuration

Definition

A tactical configuration is a triple T = (P,B, I) where

- P is a v-element set,

- B is a collection of k-element subsets of P (called ‘blocks’)with |B| = b, and

- I = {(p,B) ∈ P × B : p ∈ B} such that each element of P(called a ‘point’) belongs to exactly r blocks.

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The DSRGs from Degenerated Affine Planes

Let APl(q) denote the partial geometry obtained from AP(q) by

considering all q2 points and taking the lines of l parallel classes of

the plane.

Then APl(q) satisfies the following properties:

1 every point is incident with l lines,

2 every line is incident with q points,

3 any two points are incident with at most one line,

4 if p and L are non-incident point-line pair, there are exactlyl − 1 lines containing p which meet L.

5 APl(q) is a pg(q, l , l − 1).

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The DSRGs from Degenerated Affine Planes

Let APl(q) denote the partial geometry obtained from AP(q) by

considering all q2 points and taking the lines of l parallel classes of

the plane. Then APl(q) satisfies the following properties:

1 every point is incident with l lines,

2 every line is incident with q points,

3 any two points are incident with at most one line,

4 if p and L are non-incident point-line pair, there are exactlyl − 1 lines containing p which meet L.

5 APl(q) is a pg(q, l , l − 1).

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Construction of DSRGs from Degenerated Affine Planes

Theorem (1)

Let D = D(APl(q)) be the directed graph with its vertex set

V (D) = {(p, L) ∈ P × L : p /∈ L},

and directed edges given by

(p, L)→ (p′, L′) iff p ∈ L′.

Then D is a directed strongly regular graph with parameters:

(lq2(q − 1), lq(q − 1), lq − l + 1, (l − 1)(q − 1), lq − l + 1).

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Affine Plane of Order 2

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Antiflags

- A = (L1, p2)

- B = (L1, p4)

- C = (L2, p1)

- D = (L2, p3)

- E = (L3, p3)

- F = (L3, p4)

- G = (L4, p1)

- H = (L4, p2)

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Antiflags

- A = (L1, p2)

- B = (L1, p4)

- C = (L2, p1)

- D = (L2, p3)

- E = (L3, p3)

- F = (L3, p4)

- G = (L4, p1)

- H = (L4, p2)

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DSRG(8,4,3,1,3)

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New Graphs From Partial Geometries

The new graphs given by these constructions have parameters

(36, 12, 5, 2, 5), (54, 18, 7, 4, 7),

(72, 24, 10, 4, 10), (96, 24, 7, 3, 7),

(108, 36, 14, 8, 14), (108, 36, 15, 6, 15)

listed as feasible parameters with v ≤ 110 on “Parameters ofdirected strongly regular graphs” by S. Hobart and A. E. Brouwerat http : //homepages.cwi .nl/∼aeb/math/dsrg/dsrg .html .

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An Interesting Example Arising From Projective Planes

Let P be the point set of this projective plane.

- For a point p ∈ P, let Lp0, Lp1, . . . , Lpn denote the n + 1 linespassing through p.

- Set Bpi = Lpi − {p} for i = 0, 1, . . . , n,

- Then with B = {Bpi : p ∈ P, i ∈ {0, 1, . . . , n}}, the pair(P,B) forms a tactical configuration with parameters:

(v , b, k , r) = (n2 + n + 1, (n + 1)(n2 + n + 1), n, n(n + 1)).

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Construction of DSRG by Using Projective Planes

Theorem (2)

Let D be the directed graph with its vertex set

V = {(p,Bpi ) ∈ P × B : p ∈ P, i ∈ {0, 1, . . . , n}

and adjacency defined by

(p,Bpi )→ (q,Bqj) if and only if p ∈ Bqj .

Then D is a directed strongly regular with the parameters

(v , k, t, λ, µ) = ((n + 1)(n2 + n + 1), n(n + 1), n, n − 1, n)

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DSRG-(21,6,2,1,2) Obtained from Fano Plane

1 23, 45, 67

2 13, 46, 57

3 12, 56, 47

4 15, 26, 37

5 14, 36, 27

6 17, 35, 24

7 16, 25, 34

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DSRG-(21,6,2,1,2) Obtained from Fano Plane

1 23, 45, 67

2 13, 46, 57

3 12, 56, 47

4 15, 26, 37

5 14, 36, 27

6 17, 35, 24

7 16, 25, 34

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A General Idea

1 Consider the (ls + 1)-element setP = {1, 2, . . . , ls + 1}.

2 For each i ∈ P, letBi = {Bi1,Bi2, . . . ,Bis} be a partition ofP \ {i} into s parts (blocks) of equal sizel .

3 Let B =⋃ls+1

i=1 Bi = {Big : 1 ≤ g ≤s, 1 ≤ i ≤ ls + 1}. Then the pair (P,B)forms a tactical configuration withparameters(v , b, k, r) = (ls + 1, s(ls + 1), l , ls).

1 23, 45

2 13, 45

3 12, 45

4 12, 35

5 12, 34

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A General Idea

1 Consider the (ls + 1)-element setP = {1, 2, . . . , ls + 1}.

2 For each i ∈ P, letBi = {Bi1,Bi2, . . . ,Bis} be a partition ofP \ {i} into s parts (blocks) of equal sizel .

3 Let B =⋃ls+1

i=1 Bi = {Big : 1 ≤ g ≤s, 1 ≤ i ≤ ls + 1}. Then the pair (P,B)forms a tactical configuration withparameters(v , b, k, r) = (ls + 1, s(ls + 1), l , ls).

1 23, 45

2 13, 45

3 12, 45

4 12, 35

5 12, 34

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Theorem (3)

Let D = D(T ) be the directed graph with its vertex set

V = {(g ,B) : B ∈ Bg , g ∈ P}

and adjacency defined by

(g ,B)→ (g ′,B ′) if and only if g ∈ B ′.

Then D is a directed strongly regular graph with the parameters:

(v , k , t, λ, µ) = (ls2 + s, ls, l , l − 1, l).

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Non-Isomorphic two DSRG-(10, 4, 2, 1, 2)

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DSRG-(10, 4, 2, 1, 2)

Let l = 2 and s = 2.

- F be the set of tactical configurations.

- Consider the action of S5 on F

- T σ1 = T2 if and only if V (T1)σ = V (T2) where

V (T )σ = {(iσ, (Bij)σ) : i ∈ P, Bij ∈ B}

with natural action on Bij .

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The Block Sets of the Representatives of Seven Orbits

i B(T1) B(T2) B(T3) B(T4) B(T5) B(T6) B(T7)

1 23, 45 23, 45 23, 45 23, 45 23, 45 23, 45 23, 452 13, 45 13, 45 13, 45 14, 35 13, 45 13, 45 13, 453 12, 45 14, 25 12, 45 15, 24 14, 25 14, 25 14, 254 12, 35 12, 35 12, 35 13, 25 12, 35 12, 35 13, 255 12, 34 12, 34 13, 24 12, 34 14, 23 13, 24 14, 23

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Stabilizers and the Size of Orbits for the Action of S5 on F

T1 T2 T3 T4 T5 T6 T7D8 C2 × C2 C2 C5 o C4 C2 C2 D10

15 30 60 6 60 60 12

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New Graphs from Tactical Configuration

The new graphs obtained from tactical configurations by similarmethods have parameters:

(45, 30, 22, 19, 22), (54, 36, 26, 23, 26), (72, 48, 34, 31, 34),

(75, 60, 52, 47, 52), (81, 54, 38, 35, 38), (90, 30, 11, 8, 11),

(90, 60, 44, 38, 44), (99, 66, 46, 43, 46), (100, 40, 18, 13, 18),

(108, 36, 13, 10, 13), (108, 72, 50, 47, 50), (108, 72, 52, 46, 52),

(108, 90, 80, 74, 80)

listed as feasible parameters with v ≤ 110 on “Parameters ofdirected strongly regular graphs” by S. Hobart and A. E. Brouwerat http : //homepages.cwi .nl/∼aeb/math/dsrg/dsrg .html

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[1] Bose, R. C., Strongly regular graphs, partial geometries, andpartially balanced designs, Pacific J. Math. 13 (1963)389–419.

[2] Brouwer, A. E., Cohen, A. M. and Neumaier, A.,Distance-Regular Graphs, Springer-Verlag, Berlin, 1989.

[3] Brouwer, A. E. and Hobart, S., Directed strongly regulargraph, in: Colburn, C. and Dinitz, J. (Eds.), Handbook ofCombinatorial Designs, CRC Inc., Boca Raton, 868–875.

[4] Duval, A., A directed graph version of strongly regular graphs,Journal of Combinatorial Theory, (A) 47 (1988), 71–100.

[5] Duval, A. and Iourinski, D., Semidirect product constructionsof directed strongly regular graphs, Journal of CombinatorialTheory, (A) 104 (2003) 157-167.

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[6] Fiedler, F, Klin, M. and Muzychuk, M., Smallvertex-transitive directed strongly regular graphs, DiscreteMathematics 255 (2002) 87-115.

[7] Fiedler, F., Klin, M. and Pech, Ch., Directed strongly regulargraphs as elements of coherent algebras, in: Denecke, K. andVogel, H.-J. (Eds.), General Algebra and DiscreteMathematics, Shaker Verlag, Aachen, 1999, pp. 69–87.

[8] Godsil, C. D., Hobart, S. A. and Martin, W. J.,Representations of directed strongly regular graphs, EuropeanJournal of Combinatorics 28 (2007), no. 7, 1980–1993.

[9] Hobart, S. and Brouwer, A. E., “Parameters of directedstrongly regular graphs,” kept in A. Brouwer’s URL:[http://homepages.cwi.nl/∼aeb/math/dsrg/dsrg.html]

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[10] Hobart, S. and Shaw, T., A note on a family of directedstrongly regular graphs, European Journal of Combinatorics20 (1999), 819–820.

[11] Jørgensen, L., Search for directed strongly regular graphs.Report R-99-2016 Department of Mathematical Sciences,Aalborg University, 1999.

[12] Jørgensen, L., Directed strongly regular graphs with µ = λ,Discrete Mathematics, 231 (2001), no. 1–3, 289–293.

[13] Jørgensen, L., Non-existence of directed strongly regulargraphs, Discrete Mathematics, 264 (2003), 111–126.

[14] Klin, M., Munemasa, A., Muzychuk, M. and Zieschang, P.-H.,Directed strongly regular graphs obtained from coherentalgebras, Linear Algebra and Its Applications 377 (2004),83–109.

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[15] Klin, M., Pech, Ch. and Zieschang, P.-H., Flag algebras ofblock designs: I. Initial notions, Steiner 2-designs, andgeneralized quadrangles, preprint MATH-AL-10-1998,Technische Universitat Dresden.

[16] Olmez, O. and Song, S. Y., Construction of directed stronglyregular graphs, preprint. (arXiv:1006.5395v2 [math.CO].)

[17] Olmez, O. and Song, S. Y., Some families of directed stronglyregular graphs obtained from certain finite incidencestructures, preprint. (arXiv:1102.1491[math.CO].)

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Thanks!

THANK YOUFOR YOUR ATTENTION

Contact Sung Song: [email protected] me: [email protected]

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