Another Look at the Degree-KirchhoffIndex

JOSÉ LUIS PALACIOS, JOSÉ MIGUEL RENOMDepartamento de Cómputo Científico y Estadística, Universidad Simón Bolívar, Apartado 89000,Caracas, Venezuela

Received 23 November 2009; accepted 10 March 2010Published online 4 November 2010 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/qua.22725

ABSTRACT: Let G be an arbitrary graph with vertex set {1, 2, . . . , N} and degreesdi ≤ D, for fixed D and all i, then for the index R′(G) = ∑

i<j didjRij we show that

R′(G) ≥ 2|E|(

N − 2 + 1D + 1

).

We also show that the minimum of R′(G) over all N-vertex graphs is attained for the stargraph and its value is 2N2 − 5N + 3. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem 111:3453–3455, 2011

Key words: Kirchhoff index; star graph

1. Introduction

A mong the various indices in mathematicalChemistry, the Kirchhoff index R(G) and a

close relative of it, R′(G), that for a lack of a bettername we will call the degree-Kirchhoff index, havereceived a lot of attention in recent times. For a con-nected undirected graph G = (V, E) with vertex set{1, 2, . . . , N} and edge set E, the Kirchhoff index wasdefined by Klein and Randic [1] as

R(G) =∑i<j

Rij,

where Rij is the effective resistance of the edge ij. Werefer the reader to Refs. [2–4, 6, 7] and [9] through

Correspondence to: J. L. Palacios; e-mail: jopala@cesma.usb.ve

[15], among others, for a variety of approachesexpressing this index in terms of eigenvalues, hit-ting times of random walks, average of the Wienercapacities of its vertices, the matching number, etc.,as well as upper and lower bounds and exact val-ues for a variety of families of graphs endowed withsome form of symmetry. The degree-Kirchhof indexwas proposed by Chen and Zhang [4], defined as

R′(G) =∑i<j

didjRij,

where di is the index of the vertex i. (For all graphicaltheoretical terms the reader is referred to Ref. [5].)

The index R′(G) was also studied in Refs. [6, 7].In the latter article, through an expression in termsof eigenvalues of the transition matrix and Lagrangemultipliers, we gave the general lower bound

International Journal of Quantum Chemistry, Vol 111, 3453–3455 (2011)© 2010 Wiley Periodicals, Inc.

PALACIOS AND RENOM

R′(G) ≥ 2|E|(N − 1)2

N

for an arbitrary G. We also found an upper boundof order N5 for this index that is attained (up to theconstant of the leading term) by the barbell graph.

In this article, we want to give a new lower boundthat improves our previous one, expressed in termsof an upper bound D of the degrees of the verticesand derived entirely with electric principles. We alsoshow that the minimum value of R′(G) among all N-vertex simple connected graphs is attained by thestar graph.

2. The Bounds

Proposition 1. Let i and j be any two vertices in G.If (i, j) /∈ E then

Rij ≥ 1di

+ 1dj

; (1)

If (i, j) ∈ E then

Rij ≥ di + dj − 2didj − 1

(2)

Proof. The bound Eq. (1) is proven in Ref. [8].We will prove the bound Eq. (2). There is an edgebetween i and j. If di = 1 or dj = 1 then Rij = 1.So, we take di ≥ 2 and dj ≥ 2. Consider now allthe endpoints of all the other di − 1 edges stemmingout of i and all the dj − 1 edges stemming out of j.Short all these. Then, we get two edges in parallel:one with resistance 1 and the other with resistance

1di−1 + 1

dj−1 . Solving this into a single resistor finishes

the proof

Starting from the fact that (di − dj)2 ≥ 0, it is easy

to prove that

di + dj − 2didj − 1

≥ 1di + 1

+ 1dj + 1

(3)

and thus our bound for (i, j) ∈ E improves the resultin Ref. [8]. Notice that both expressions in Eq. (3)coincide when di = dj. Notice also that our bound inEq. (2) is valid when either di = 1 or dj = 1.

Now we can prove

Proposition 2. For any N-vertex graph, the degreesof whose vertices 1 ≤ i ≤ N satisfy di ≤ D for fixed Dand all i, we have

R′(G) ≥ 2|E|(

N − 2 + 1D + 1

). (4)

Proof. Using Eqs. (1) and (2) we can write

R′(G) =∑i<j

didjRij ≥∑i<j

d(i,j)=1

didjdi + dj − 2

didj − 1

+∑i<j

d(i,j)≥2

didj

(1di

+ 1dj

)

=∑i<j

(di + dj) −∑i<j

d(i,j)=1

2 +∑i<j

d(i,j)=1

di + dj − 2didj − 1

= 2|E|(N − 2) +∑i<j

d(i,j)=1

di + dj − 2didj − 1

. (5)

To bound the last term it is enough to consider the

real function f (x) = x+dj−2

xdj−1 in the interval [1, D] and

notice that it is decreasing in that interval, so that

f (x) ≥ f (D) = D+dj−2

Ddj−1 . A similar argument with the

function g(x) = D+x−2Dx−1 shows that g also attains its

minimum at x = D and therefore

∑i<j

d(i,j)=1

di + dj − 2didj − 1

≥ 2|E| 1D + 1

. (6)

Inserting Eq. (6) into Eq. (5) finishes the proof

The new bound in Eq. (4) and the previous bound2|E| (N−1)2

N coincide whenever D = N − 1, so in par-ticular they both give the exact value for R′(KN) =(N − 1)3. For all other cases, when D < N − 1, thenew bound is better.

The value (N − 1)3 for R′(KN) is not minimal. Wewill show now that the minimum value of R′(G) isattained for the star graph SN , for which R′(SN) =2N2 − 5N + 3. First we prove the following

Proposition 3. For any tree T we have

R′(T) ≥ R′(SN) = 2N2 − 5N + 3 (7)

3454 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 111, NO. 14

KIRCHHOFF INDEX

Proof. For any tree T, using Eq. (1) , we have

R′(T) ≥∑i<j

d(i,j)=1

didj +∑i<j

d(i,j)≥2

didj

(1di

+ 1dj

)

=∑i<j

(di + dj) +∑i<j

d(i,j)=1

(didj − di − dj)

= 2(N − 1)2 +∑i<j

d(i,j)=1

(didj − di − dj).

Now, we minimize the last summation: the sum-mand didj − di − dj is nonnegative if and only if bothdi ≥ 2 and dj ≥ 2. Otherwise, if either di = 1 ordj = 1 the summand equals −1. In other words, theminimum is attained when we have that for eachedge one of its vertices is a leaf, in which case wehave the star graph, and the value of the boundbecomes

2(N − 1)2 − (N − 1) = (N − 1)(2N − 3) = R′(SN)

Finally, we prove

Proposition 4. For any graph G we have

R′(G) ≥ R′(SN) = 2N2 − 5N + 3 (8)

Proof. If G is not a tree, then |E| ≥ N, andtherefore, by proposition 2 we have

R′(G) ≥ 2N(

N − 2 + 1D + 1

)> 2N2 − 5N + 3

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