Broder and Karlin’s Formula for HittingTimes and the Kirchhoff Index

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

Received 10 June 2009; accepted 15 June 2009Published online 25 November 2009 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/qua.22396

ABSTRACT: We give an elementary proof of an extension of Broder and Karlin’sformula for the hitting times of an arbitrary ergodic Markov chain. Using this formulain the particular case of random walks on graphs, we give upper and tight lowerbounds for the Kirchhoff index of any N- vertex graph in terms of N and its maximaland minimal degrees. We also apply the formula to a closely related index that takesinto account the degrees of the vertices between which the effective resistances arecomputed. We give an upper bound for this alternative index and show that the boundis attained—up to a constant—for the barbell graph. © 2009 Wiley Periodicals, Inc. Int JQuantum Chem 111: 35–39, 2011

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 V � {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 pastfew 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 (refer [2–9],among others).

Recently, upper and lower bounds for this indexhave been given in general contexts. Zhou and Tri-najstic gave in [10] tight lower bounds for R(G) andarbitrary G, working with the characterization ofthe Kirchhoff index as

R�G� � N �i�1

N�1 1�i

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

International Journal of Quantum Chemistry, Vol 111, 35–39 (2011)© 2009 Wiley Periodicals, Inc.

where the �is are the eigenvalues of the Laplacianmatrix of G. They also found in [11] upper andlower bounds in terms of the largest and smallerdegrees of G, working with the eigenvalues of thenormalized Laplacian. We found similar bounds([12, 13]) for the particular case of regular graphsworking from the viewpoint of a random walk onthe graph G, and the characterization of the Kirch-hoff index as

R�G� �1

2�E��i, j

EiTj (2)

where EiTj is the expected number of jumps, start-ing from vertex i, that the random walk needs toreach the vertex j. This characterization is based onthe fact (refer [14]) that the so called commute timescan be expressed, for any i and j, as follows:

EiTj � EjTi � 2�E�Rij . (3)

In this article, we extend our previous results forregular graphs to arbitrary graphs, extending aswell some of the results of Zhou and Trinajstic(2009). Once again, we use the probabilistic ap-proach to the Kirchhoff index via random walks ongraphs.

2. The Formulas

Broder and Karlin proved in [15] that for anergodic aperiodic reversible Markov chain (or inother words, for a random walk on a nonbipartitegraph) the following equation holds

�i, j

wiwjEiTj � �j�2

11 � �j

(4)

where the �js are the eigenvalues of the transitionprobability matrix P of the chain, and w � [w1, . . . ,wN] is its stationary distribution, that is, the uniquerow vector satisfying wP � w.

More recently, Aldous and Fill showed in [14],by embedding the Markov chain into a continuousprocess, that

�j

wjEiTj � �j�2

11 � �j

, (5)

holds regardless of the value of i, for a generalergodic Markov chain, from which (4) follows eas-ily. In the following proposition we will give acomplete proof of this fact, for the general case ofergodic chains and using only simple discrete ar-guments, with the added bonus of the connectionwith the so called fundamental matrix Z defined as

Z � �I � P � W��1,

where I is the identity matrix, and W is the matrixall of whose rows are identical to w.

Proposition 1. For any ergodic Markow chain wehave

�i, j

wiwjEiTj � tr�Z� � 1 � �j�2

11 � �j

. (6)

Proof. The fact that

EiTj �Zjj � Zij

wj(7)

holds, for any i and j, is undergraduate material(refer [16] for instance). Recalling that Zc � c, wherec � [1, 1,. . . , 1]T and T means “transpose,” afterrearranging and adding we get

�j

wjEiTj � tr�Z� � 1, (8)

which shows that the left hand side of (8) is aconstant independent of i, sometimes referred to asKemeny’s constant, (refer [16]). Multiplying by wi

and adding over i, we get

�i, j

wiwjEiTj � tr�Z� � 1 (9)

Here tr(Z) stands for the trace of Z. If the eigen-values of the transition probability matrix P of theergodic chain are the (possibly complex) numbers1 � �1, �2, . . . , �N, then ��j� �1 for 2 � j � N � 1 and��N� � 1 with equality if and only if the chain isperiodic, in which case �N � �1. From the defini-tion of Z we have

zjj � 1 � �m�1

�

�Pjjm � wj�. (10)

Adding over j we get

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36 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 111, NO. 1

tr�Z� � 1 � N � 1 � �m�1

�

�tr�Pm� � 1� � N � 1

� �m�1

�

��2m � �3

m � · · · � �Nm�. (11)

Assume first that the chain is aperiodic. Then, allthe geometric series in (11) converge and we canwrite

tr�Z� � 1 � N � 1 � �j�2

�j

1 � �j� �

j�2

11 � �j

. (12)

If the chain is periodic, then �N � �1, and thusthe last series in (11) does not converge. To avoidthis problem, we consider the modified chain withtransition probability P�1/2 (I�P) , where I is theidentity matrix. This amounts to having a “holdingtime” in each state before jumping to the next, andmaking the modified chain aperiodic. Then (12)holds for the modified chain, that is

tr�Z� � 1 � �j�2

11 � �j

� 2�j�2

11 � �j

. (13)

But,

Z � �I � P � W��1 � �12I �

12P � W��1

, (14)

and since WP � PW � W2 � ZW � W, then

�2Z�W��12I �

12P � W� � Z�I � P � W� � I.

Therefore,

Z � 2Z � W,

and

tr(Z)�2tr(Z)�tr(W)�2tr(Z) � 1.

Inserting this latter equation into (13) finishes theproof.

Chen and Zhang introduced in [7] the followingindex, closely related to the Kirchhoff index:

R��G� � �i, j

didjRij,

where di is the index of the vertex i. (For all graphtheoretical terms the reader is referred to [17]). Be-cause the stationary distribution of the randomwalk on G is given by wj � dj/2�E� , it is clear that(3) and (6) imply the following,

Corollary 1. For an arbitrary G we have

R��G� � 2�E��tr(Z� � 1) � 2�E��j�2

11 � �j

.

(15)

In what follows, � and will be, respectively, thesmallest and the largest degrees among all verticesof G. We will now get some bounds in terms ofthese two numbers.

Proposition 2. For any G � �V,E� we have

N

�tr�Z� � 1� � R�G� �N�

�tr�Z� � 1�. (16)

Proof. From (7), adding over all possible valueswe get

�i, j

EiTj � �i, j

Zjj � Zij

wj.

In the case of a random walk on the graph G, wecan write the above equality as

R�G� � �i, j

Zjj � Zij

dj(17)

which immediately implies

N� �

j

Zjj � 1� � R�G� �N� � �

j

Zjj � 1� �

Corollary 2. For an arbitrary G we have

N�

j�2

11 � �j

� R�G� �N��j�2

11 � �j

.

Proof. Use (6) and (16).

Corollary 3. For an arbitrary G we have

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VOL. 111, NO. 1 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 37

�N � 1�2

� R�G�.

If the graph is bipartite, the bound can be improved to

N�2N � 3�

2� R�G�.

Proof. If A is the incidence matrix of G and D thediagonal matrix whose diagonal entries are the de-grees of the vertices of G, then the matrices P � Dand D�1/2AD�1/2 are similar, and the latter beingsymmetric implies that its eigenvalues, and hencethose of P, are real numbers.

The method of Lagrange multipliers applied tothe real function

f� x2, · · ·, xN� � �j�2

N 11 � 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 and its value is �N � 1�2/N . Using this inthe left inequality in corollary 2 finishes the proof inthe general case. For the bipartite graph, since xN ��1, use again Lagrange multipliers on the function

f� x2,· · ·, xN�1� �12 � �

j�2

N�1 11 � xj

subject to the condition x2 � . . . � xN�1 � 0. Nowthe minimum is attained at x2 � . . . � xN�1 � 0 andits value is 2N�3/2.

Corollary 4. For an arbitrary G we have

2�E��N � 1�2

N � R��G�.

If the graph is bipartite, the bound can be improved to

�E��2N � 3� � R��G�.

Proof. Use the arguments in corollary 3 to min-imize �j�21/1��j and (15).

Corollary 5. If �1, , �N are the eigenvalues of thenormalized Laplacian L of an arbitrary G then

N�

j�2

1�j

� R�G� �N��j�2

1�j

.

Proof. With the notation in the proof of corollary3, the ordinary Laplacian is L � D � A and thenormalized Laplacian is L � D�1/2LD�1/2 � I �D�1/2 AD�1/2. But the matrices P � D and D�1/2

AD�1/2 are similar and so they have the sameeigenvalues, and therefore, 1 � �j � �j. Now applycorollary 1.

Corollary 6. For an arbitrary G we have

N2�E�R��G� � R�G� �

N2��E�R��G�. (18)

If G is d-regular then

R� � d2R.

Proof. Use (15) and (16).

Corollaries 2 and 3 generalize our results ob-tained in [12, 13] for regular graphs. Corollaries 3and 4 improve results of Zhou and Trinajstic(2009) who obtained the lower bounds for bipar-tite graphs and showed that they are attained forcomplete bipartite graphs. Our general lowerbounds are attained for KN. Corollaries 5 and 6were obtained by Zhou and Trinajstic (2009)with different methods.

We finally turn our attention to the maximalvalue of R�(G) among all N-vertex graphs. Weproved in [8] that the maximal value of R(G) isobtained for the linear graph and it is of order1/6N3. Thus it is clear that the order of R�(G) forthe linear graph is 2/3N3, but this is hardly op-timal. First, we prove a general upper bound:

Corollary 7. For an arbitrary G we have

R��G� �N5

6 . (19)

Proof. From (18), we get

R��G� � 2R�G� �N5

6 .

Consider now the (1/3, 1/3, 1/3)-barbell graphwhich consists of two copies of KN/3 attached at theendpoints of a linear graph on N/3 vertices. Then

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38 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 111, NO. 1

�E� � 2N2/9 and �2 � 1 � c/N3 where c � 54� O�1/N�, as was shown in [18], and thus by (15):

R��G� � 2�E��j�2

11 � �j

�2

243N5,

and therefore the (1/3, 1/3, 1/3)-barbell attains thevalue of the upper bound in (19) up to a constant.We conjecture that this graph in fact attains thelargest value of R�(G) among all N-vertex graphs.

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