Bounds for the Kirchhoff Index ofRegular Graphs via the Spectra of TheirRandom Walks

JOSE LUIS PALACIOS, JOSE MIGUEL RENOMDepartamento de Computo Científico y Estadística, Universidad Simon Bolívar, Apartado 89000,Caracas, Venezuela

Received 30 March 2009; accepted 21 April 2009Published online 13 October 2009 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.22323

ABSTRACT: Using probabilistic tools, we give tight upper and lower bounds for theKirchhoff index of any d-regular N-vertex graph in terms of d, N, and the spectral gapof the transition probability matrix associated to the random walk on the graph. Wethen use bounds of the spectral gap of more specialized graphs, available in theliterature, in order to obtain upper bounds for the Kirchhoff index of these specializedgraphs. As a byproduct, we obtain a closed-form formula for the Kirchhoff index of thed-dimensional cube in terms of the first inverse moment of a positive binomial variable.© 2009 Wiley Periodicals, Inc. Int J Quantum Chem 110: 1637–1641, 2010

Key words: hitting times; fundamental matrix; Kemeny’s constant

1. Introduction

T he Kirchhoff index R(G) of a connected undi-rected graph G � (V, E) with vertex set {1,

2,. . . , N} and edge set E was defined by Klein andRandic [1] as

R�G� � �i�j

Rij,

where Rij is the effective resistance of the edge ij.This index has been studied intensely in the past

few years from a variety of viewpoints: graph the-ory, study of the Laplacian and of the normalizedLaplacian, electric networks and probabilistic argu-ments for hitting times, etc., and its value has beendetermined for a variety of families of graphs en-dowed with some form of symmetry (see [2–8]among others).

Recently, upper and lower bounds for this indexhave been given in more general contexts. Zhouand Trinajstic gave in [9] tight lower bounds forR(G) and arbitrary G, working with the character-ization of the Kirchhoff index as

R�G� � N �i�1

N�1 1�i

, (1)Correspondence to: J. L. Palacios; e-mail: jopala@cesma.usb.veor jopala@usb.ve

International Journal of Quantum Chemistry, Vol 110, 1637–1641 (2010)© 2009 Wiley Periodicals, Inc.

where the �is are the eigenvalues of the Laplacianmatrix of G. In the case of d-regular graphs, whereall vertices have exactly d neighbors, (see [10] for allthe graph-theoretical notions used here), their re-sult reads as:

N1 � d �

N�N � 2�2

Nd � 1 � d � R�G�. (2)

For this class of d-regular graphs, Palacios found in[11] the bounds

N�N � 1�

2d � R�G� �3N3

d , (3)

working from the viewpoint of a random walk onthe graph G, and the characterization of the Kirch-hoff index as

R�G� �1d�

j

E1Tj �Nd �tr�Z� � 1�, (4)

where E1Tj is the expected number of jumps, start-ing from vertex 1, that the random walk needs toreach the vertex j, and tr(Z) is the trace of theso-called fundamental matrix Z obtained as fol-lows: if P is the transition probability matrix of thewalk and W is the matrix all whose rows are iden-tical to the stationary distribution w (which is theunique row vector satisfying wP � w), then Z �(I � P � W)�1.

In fact, the derivation of the bounds in (3) de-pended exclusively on the interpretation of R(G) asa sum of expected hitting times and did not use theequality R�G� � N/d (tr (Z) � 1), which resembles(1) except that in (4) we have the sum of the eigen-values of Z and not of their inverses. It is thepurpose of this note to focus on this latter equalityin order to obtain new tight upper and lowerbounds for R(G) in case G is connected and d-regular, which will be the only restrictions on thegraph in the next section.

2. The Bounds

The constant K � tr(Z) � 1 is a well-knownquantity in Markov Chain Theory called Kemeny’sconstant. If the eigenvalues of the transition prob-ability matrix P of the random walk on G are

1 � �1 � �2 � �3 � · · · � �N � � 1, (5)

we know that these numbers are all real because Pis symmetric, and we also know that they are re-lated to the eigenvalues �i of the incidence matrix Aby the relation d�i � �i. We will work with the �isand P and get bounds for K, following [12] andactually extending their result to the case where thechain is only irreducible and reversible (and notnecessarily aperiodic), in order to prove our

Proposition 1. For any d-regular graph G on N ver-tices, we have

�N � 1�2

d � R�G� �N�N � 1�

d�1 � �2�. (6)

If the graph is bipartite, the lower bound can be improvedslightly to

N�2N � 3�

2d � R�G�. (7)

Proof. Let Z � (zij) be as in the introduction. Fromthe definition, we have

zkj � kj � �m�1

�

�Pkjm � wj�, (8)

where kj is the Kronecker delta and 1 � j, k � N.We also have that

zjj � 1 � �m�1

�

�Pjjm � wj�. (9)

Subtracting (8) from (9) and adding over j, we get

K � tr�Z� � 1 � N � 1 � �m�1

�

�tr�Pm� � 1�

� N � 1 � �m�1

�

��2m � �3

m � · · · � �Nm�. (10)

Assume first that the graph is not bipartite. Thenthe random walk on it is an irreducible and aperi-odic Markov chain, and the rightmost inequality in(5) is strict. Therefore, all the geometric series in (10)converge and we can write

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1638 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 110, NO. 9

K � N � 1 � �j�2

N�j

1 � �j. (11)

The method of Lagrange multipliers applied tothe function

f� x2,· · ·, xN� � �j�2

N xj

1 � xj

subject to the condition 1 � x2. . . � xN � 0 on thedomain 1 � x2 � ��� � xN � �1 tells us that theminimum of the function is attained at x2 � · · ·� xN � � 1/N � 1 . This conclusion plus thefact that the real function f�x� � x/1 � x is in-creasing in the interval ( � �, 1), allow us to write

�N � 1�2

N � K �N � 11 � �2

. (12)

If we use (12) in (4), we obtain the desired result.If the graph is bipartite, then the random walk on

it is a periodic Markov chain and the smallest eig-envalue is �N � �1, and thus the last series in (10)does not converge. To circumvent this problem, weconsider the modified walk with transition proba-bility P � 1/2 �I � P�, where I is the identitymatrix. This amounts to having a “holding time” ineach vertex before jumping to the next and makingthe modified chain aperiodic. Then (12) holds forthe modified chain, so in particular

K �N � 11 � �2

. (13)

It is easy to see now that �i is an eigenvalue of Pif and only if 2�i � 1 is an eigenvalue of P, andsince the real function g(x) � 2x � 1 is increasing,we have that

�2 �12��2 � 1�. (14)

Also, if we consider the characterization of Ke-meny’s constant as

K �1N�

j

E1Tj,

where Tj is the hitting time of the vertex j for themodified walk, then we can express K in terms of Kas follows: the modified random walk amounts to awalk on the graph modified to include a loop inevery vertex with conductance d while all otherconductances on the edges are equal to 1. Now, wecan express every expected hitting time in electricfashion

EaTb �12�

x

C� x�Rab � Rbx � Rax,

as was proven in [13]. Here Rvw and C�x� denote,respectively, the effective resistance between verti-ces v and w, and the sum of all conductances ema-nating from x in the modified graph. But the loopshave no effect on the computation of the effectiveresistances, so that Rvw � Rvw and C�x� � 2d� 2C�x� Rvw and C(x) are computed in the originalgraph), and therefore E1Tj � 2E1Tj and

K � 2K. (15)

Therefore, substituting (14) and (15) into (13), wesee that the right hand side of (12) holds for thebipartite graph. Moreover,

R�G� �Nd

K �N2d2K �

NKd � R�G�,

and so we can compute R(G) using the eigenvalues�i of G like in (10):

R�G� �Nd �N � 1 � �

j�2

N�j

1 � �j�

�N2d�N � 1 � �

j�2

N 1 � �j

1 � �j� . (16)

Now we notice that the summand for j � N is zero,and applying Lagrange multipliers to the function

f�x2,· · ·,xN�1� � �j�2

N�11 � xj/1 � xj subject to the

condition x2 � ��� � xN � 1 � 0, we get (7).We notice first that we have extended the result

in [12] to the case of reversible periodic chains. Wealso notice that the lower bounds in (6) and (7) arebetter than that of (3) by a factor of 2, much in thesame way as the lower bound in (2). The expression(16) is similar to (1) though not quite equivalent.

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The quantity 1 � �2 appearing in the upper boundis usually called the spectral gap and is relevant ina number of contexts such as expander graphs andthe speed of convergence of the transition probabil-ities of a Markov chain to its stationary distribution.

The rest of this section is dedicated to someexamples.

1. The complete graph KN. Its spectrum consists of1 and � 1/N � 1 with multiplicity N � 1. Ourbounds become

N � 1 � R�KN� � N � 1

which are obviously tight, since R(KN) � N � 1.2. The complete bipartite graph. For KN,N, the spec-

trum is 1, �1, and 0 with multiplicity 2N � 2, andtherefore our bounds become

4N � 3 � R�G� � 2�2N � 1�,

which are tight including the leading term constant.In fact, a direct application of (16) shows thatR(G) � 4N � 3.

3. The N-cycle. Now the spectrum is the set�cos�2i/N��, 0 � i � N � 1, so that �2

� cos�2/N�. The bounds become

�N � 1�2

2 � R�G� �N�N � 1�

2�1 � cos�2

N ��,

in case N is odd, with the lower bound becomingN�2N � 3�/4 in case N is even (bipartite graph).Here, both these bounds give the exact value forN � 3, but they are not very good asymptotically,because for N large the approximation cos x 1 � x2/ 2 shows that the bounds are away fromthe actual value (see [3]), 1/12�N3 � N�, by afactor of N.

4. The d-dimensional cube. Here, the spectrum is

the set �d � 2i/d�, 0 � i � d, with multiplicity �di�,

so the bounds become

2d�1�22d�1 � 3�

d � R�G� � 2d�1�2d � 1�.

In this case, we can get a closed-form formula forR(G) using the expression (16):

R�G� �2d

2d�2d � 1 � �i�1

d�1�di �d � i

i � �2d�1

d �2d � 1

� d�2d�1 � 1� E1X� , (17)

where X is a positive binomial variable with param-eters d � 1 and 1/2. There is no closed-form for-mula for E1/X, but at least half a century of ap-proximations, of which we choose the simple one in[14]: E1/X � 2�d � 3�/�d � 1��d � 4� that, afterinserting in (17) allows us to write R�G� � 22d/d.

Thus, we see that in this case, the upper boundfurnished by the spectrum is off by a logarithmicfactor of the number of vertices, whereas the lowerbound is correct including the constant of the lead-ing term.

3. Further Bounds

The new bounds we have given in the previoussection, based on the spectrum, perform better thanthe ones in (3) in most of our examples, though theyare not universally better. Still, from our upperbound, we can deduce a number of additionalbounds in more specialized cases using the abun-dant literature concerning the spectral gap. For ex-ample, we can quote the following propositions,which arise from corresponding spectral bounds in[15] (pp. 56, 112 and 113), providing us with boundsthat potentially can be of orders as low as N or N2:

Proposition 2. Suppose there is a set P of �N2� paths

joining all pairs of vertices such that each path in P haslength at most L and each edge is contained in at most Mpaths in P. Then

R�G� � �N � 1� ML.

Proposition 3. Suppose G is edge-transitive with di-ameter D. Then

R�G� �N�N � 1� D2

d

Proposition 4. Suppose G is vertex-transitive withdiameter D. Then

R�G� � N�N � 1� D2.

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Examples. (i) Let G be a geodetic d-regular graphwith diameter 2. Then in Proposition 2, L � 2,because for every pair of vertices, either they areneighbors or they are at distance 2 joined by aunique common neighbor. Also, given the edge vw,the vertex v can act as a connector between w and atmost d � 1 neighbors of v, and likewise the vertexw can act as a connector between v and at most d �1 neighbors of w, therefore the edge vw is used in atmost 2(d � 1) paths and the bound for these graphsbecomes

R�G� � 4�d � 1��N � 1�,

which seems to be a good one, since it was provenin [8] that for d-regular geodetic graphs with diam-eter 2 and no triangles, R(G) � N � 1 � d(N � 2).

(ii) It was shown in [7] that for G d-regular anddistance-transitive, with opposite vertices at dis-tance 2, R�G� � N � 1 � N�N � d � 1�/d.Proposition 3 deals with a less restrictive and thuslarger class of graphs, and the case D � 2 showsthat the bound can be rather tight.

(iii) The order of the bound in proposition 4 canbe as bad as N4 (useless) or as good as N2 for small

diameters. This latter case occurs for many classesof Cayley graphs (see for example [16]).

References

1. Klein, D. J.; Randic, M. J Math Chem 1993, 12, 81.2. Klein, D. J.; Palacios, J. L.; Randic, M. J.; Trinajstic, N. J Chem

Inf Comput Sci 2004, 44, 1521.3. Lukovits, I.; Nikolic, S.; Trinajstic, N. Int J Quantum Chem

1999, 71, 217.4. Lukovits, I.; Nikolic, S.; Trinajstic, N. Croat Chem Acta 2000,

73, 957.5. Yang, Y.; Zhang, H. Int J Quantum Chem 2007, 108, 503.6. Chen, H.; Zhang, F. Discrete Appl Math 2007, 155, 654.7. Palacios, J. L. Int J Quantum Chem 2001, 81, 29.8. Palacios, J. L. Int J Quantum Chem 2001, 81, 135.9. Zhou, B.; Trinajstic, N. Chem Phys Lett 2008, 455, 120.

10. Wilson, R. J. Introduction to Graph Theory; Oliver & Boyd:Edinburgh, 1972.

11. Palacios, J. L. Technical report. Available at http://www.cesma.usb.ve/wiki/index.php/Portada.

12. Levene, M.; Loizou, G. Am Math Mon 2000, 109, 741.13. Tetali, P. J Theor Prob 1991, 4, 101.14. Mendenhall, W.; Lehman, E. H. Technometrics 1960, 2, 227.15. Chung, F. R. K. Spectral Graph Theory; American Mathe-

matical Society: Providence, RI, 1997.16. Curtin, E. Discrete Math Theor Comput Sci 2001, 4, 123.

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