Statistics and Probability Letters 82 (2012) 783–785

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Statistics and Probability Letters

journal homepage: www.elsevier.com/locate/stapro

On partial sums of hitting timesJosé Luis Palacios ∗, José M. RenomUniversidad Simón Bolívar, Caracas, Venezuela

a r t i c l e i n f o

Article history:Received 8 October 2011Received in revised form 13 December 2011Accepted 14 December 2011Available online 16 January 2012

Keywords:Effective resistanceKirchhoff index

a b s t r a c t

We conjecture that if Tj is the hitting time of vertex j thenj

EiTj ≥ (N − 1)2,

for all i, for a randomwalk on any connected graph G = (V , E) with |E| = N . We prove theconjecture for a family of graphs containing the regular graphs and obtain slightly betterbounds for trees and non-regular edge-transitive graphs.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

On a connected undirected graph without loops G = (V , E) and vertex set V = {1, . . . ,N} we can define the simplerandom walk (SRW) on G as the Markov chain Xn, n ≥ 0, that from its current vertex v jumps to a neighboring vertex wwith probability pvw = 1/d(v), where d(v) is the number of neighbors of v. This amounts to endowing the chain with theN × N transition probability matrix P = DA, where D is a diagonal matrix whose non-zero entries are Dii =

1d(i) and A is the

adjacency matrix of the graph, with entries Aij = 1 in case (i, j) ∈ E and 0 otherwise. The hitting time (or first passage time)Tb of the vertex b is the number of jumps that the walk takes until it lands on b, and its expected value when the walk startsat a is denoted by EaTb.

Global sums of hitting times have received a good deal of attention, as attested by the recent articles by Tejedor et al.(2009) and Zhang et al. (2010), and references therein. These global sums are either of the form

1N(N − 1)

i=j

EiTj, (1)

or they are defined averaging the expected hitting times not with the equal weights 1N(N−1) , but with their respective

stationary distributions:

B(G) =

i=j

πiπjEiTj,

where π = [π1, . . . , πN ] is the unique probabilistic row vector satisfying πP = π . (In the case of random walks on graphsit is readily seen that πi =

d(i)2|E|

.)Partial sums of hitting times are also of interest, and they are found in relation to cover times (Broder and Karlin, 1989;

Coppersmith et al., 1996). Also, sums of hitting times, either global or partial, are often found in theMathematical Chemistry

∗ Correspondence to: Departamento de Cómputo Científico y Estadística, Universidad Simón Bolívar, Apartado 89,000, Caracas, Venezuela. Tel.: +58 2129413796; fax: +58 212 9063234.

E-mail address: jopalal@gmail.com (J.L. Palacios).

0167-7152/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2011.12.012

784 J.L. Palacios, J.M. Renom / Statistics and Probability Letters 82 (2012) 783–785

literature, specifically in relation to the Kirchhoff index and similar indices, where the relationship between random walkson graphs and electric networks is exploited (Palacios and Renom, 2010, 2011). The Kirchhoff index of G is defined asR(G) =

i<j Rij, where Rij is the effective resistance, as computed by Ohm’s law, between vertices i and j.

A well-known partial sum of hitting times that has a closed form expression is the so-called Kemeny’s constant:j

πjEiTj =

j

11 − λj

, (2)

which turns out to be independent of i, and where the λjs are the eigenvalues of the transition probability matrix. Eq. (2) isalso known as the random target lemma (see the online publications of Aldous and Fill, Hunter).

We turn our attention in this note to the partial sums

F(i) =

j

EiTj,

that in general depend on i and apparently do not have a closed form expression in terms of eigenvalues such as (2). Weconjecture that for any graph G, F(i) ≥ (N − 1)2, for all i, and prove the conjecture for a family of graphs that includes theregular graphs, as well as for trees and edge-transitive graphs, for which we find slightly better lower bounds.

2. The bounds

A first approximation is given by the following proposition, proven in Palacios (2010) that we include here forcompleteness.

Proposition 1. For SRW on any finite graph with N vertices we have

F(i) ≥

N2

, (3)

for any i.

Proof. Let S(j) be the first time n such that the set of vertices thewalk has visited by time n, {X0, X1 . . . , Xn}, contains exactlyj different vertices. Then S(j) ≥ j − 1 and S(1) = 0. Also,

Nj=1

Tj =

Nj=1

S(j).

Taking expected values in the previous equation we obtainNj=1

EiTj =

Nj=1

EiS(j) ≥

Nj=1

(j − 1) =

N2

. �

Thus we know that the order of F(i) is at least N2, although the constant 12 is probably not optimal. The next proposition

proves the conjecture for a large family of graphs.

Proposition 2. For SRW on any finite graph with N vertices we have

F(i) ≥2|E|

∆

(N − 1)2

N, (4)

where ∆ is the largest degree in the graph.

Proof. For any i, we use the random target lemma and obtainj

EiTj ≥2|E|

∆

j

πjEiTj =2|E|

∆

j

11 − λj

≥2|E|

∆

(N − 1)2j(1 − λj)

=2|E|

∆

(N − 1)2

N,

where the last inequality is a consequence of the Cauchy–Schwarz inequality. �

This result implies that the conjecture is true for any graph satisfying 2|E|

∆N ≥ 1, which happens for ∆-regular graphs, amongothers. The lower bound (N−1)2 is attained in the complete graph. This proposition, obviously, does not prove the conjecturefor trees, but for them we obtain a larger lower bound.

J.L. Palacios, J.M. Renom / Statistics and Probability Letters 82 (2012) 783–785 785

Proposition 3. For SRW on any tree with N vertices we have

F(i) ≥ (N − 1)(3N − 4)/2. (5)

Proof. We use a compact recent formula for EiTj in trees given by Bapat (2011):

EiTj = αj − αi + (N − 1)dij, (6)

where dij is the distance between vertices i and j and αj =

i dij. Adding (6) over all jwe get

F(i) =

j

αj − Nαi + (N − 1)

j

dij =

j

αj − αi = 2i<j

dij − αi. (7)

It is plain to see that in a tree the effective resistance Rij, between vertices i and j, is precisely dij, so Eq. (7) indeed showsthat

F(i) = 2i<j

Rij − αi = 2R(G) − αi.

The largest αi can get for any i for any tree is

N2

in the case of the linear graph, when i is one of the two leaves, so that

F(i) ≥ 2R(G) −

N2

.

Now it is well known (Dobrynin et al., 2001) that R(G) attains its minimum among trees in the case of the star tree, for whichR(G) = (N − 1)2, and we are done. �

We end with a proof of the conjecture for all edge transitive graphs. A graph is edge transitive if for every pair of edges(considered as undirected edges), there is a graph automorphism mapping one edge onto the other. If a graph is edgetransitive it is not necessarily vertex transitive, not even regular, but as we showed in Palacios and Renom (1998) we havethat

Proposition 4. If G = (V , E) is edge transitive and regular, then EiTj = N − 1 whenever (i, j) ∈ E. If G is not regular thenthere are only two distinct degrees ∆1 and ∆2,G is bipartite with partition Vi, i = 1, 2, where Vi is the set of nodes with degree∆i, i = 1, 2, and |V1|∆1 = |V2|∆2 = |E|. Also, if (i, j) ∈ E, EiTj = 2|V2| − 1 when i ∈ V1, j ∈ V2 and EiTj = 2|V1| − 1 wheni ∈ V2, j ∈ V1.

Proposition 5. The conjecture is true for SRW on any edge transitive graph with N vertices.

Proof. In view of Proposition 2, we need to prove the result only when the graph is not regular.If that is the case then the graph is bipartite. Starting from i ∈ V1, the walk must reach all vertices in V2, which for each

vertex takes at least in average 2|V2| − 1 steps, and the walk must also reach |V1| − 1 vertices in V1, which for each vertextakes at least in average 2|V1| steps. Therefore

F(i) ≥ 2N2− (4|V1| |V2| + |V2| + 2|V1|).

A similar expression, exchanging the roles of V1 and V2 is obtained when i ∈ V2. These expressions are minimized when|V1| = |V2| = N/2 and in this case the lower bound (which is attained by the complete bipartite graph KN/2,N/2) becomesN2

−32N ≥ (N − 1)2. �

References

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