On the Probabilistic Representation of the Free Effective Resistance of Infinite Graphs

TL;DR

This paper characterizes the conditions under which the free effective resistance of infinite graphs can be represented using simple hitting probabilities of the graph's random walk, providing a clear link between resistance and probabilistic behavior.

Contribution

It offers a complete characterization of when free effective resistance can be expressed via hitting probabilities in infinite graphs, connecting electrical and probabilistic graph properties.

Findings

Identifies conditions for resistance representation in terms of hitting probabilities

Provides a complete characterization for infinite graphs

Bridges electrical network theory with probabilistic analysis

Abstract

We completely characterize when the free effective resistance of an infinite graph can be expressed in terms of simple hitting probabilities of the graphs random walk.

Equations

$$c_x:=\sum _{y\in V}c(x,y)<\infty .$$

$$p(x,y)=\frac{c(x,y)}{c_x},x,y\in V$$

$${\tau }_A$$

$${\tau }_A^{+}$$

$$R(x,y)$$

$$=\frac{1}{c_x\cdot {\mathbb{P}}_x[{\tau }_y\le {\tau }_x^{+}]}$$

$$=\frac{1}{c_x\cdot {\mathbb{P}}_x[{\tau }_y<{\tau }_x^{+}]}.$$

$$\frac{1}{c_x\cdot {\mathbb{P}}_x[{\tau }_y\le {\tau }_x^{+}]}\le R^F(x,y)\le \frac{1}{c_x\cdot {\mathbb{P}}_x[{\tau }_x<{\tau }_x^{+}]}$$

$$R^F(x,y)=\frac{1}{c_x\cdot {\mathbb{P}}_x[{\tau }_y<{\tau }_x^{+}]}$$

$$R^F(x,y)=\underset{n\to \infty }{\lim }{R}_{G_n}(x,y).$$

$$p_n(x,y)=\frac{c_n(x,y)}{(c_n{)}_x}=\frac{c(x,y)}{{\sum }_{w\in V_n}c(x,w)}$$

$$p_n(x,y)=p(x,y)\cdot \frac{c_x}{(c_n{)}_x}=p(x,y)\cdot \left(1+\frac{{\sum }_{v\notin V_n}c(x,v)}{{\sum }_{v\in V_n}c(x,v)}\right)\ge p(x,y).$$

$$R_{G_n}(x,y)=\frac{1}{(c_n{)}_x{\mathbb{P}}_x^n[{\tau }_y<{\tau }_x^{+}]}$$

$$\underset{n\to \infty }{\lim }{\mathbb{P}}_x^n[{\tau }_y<{\tau }_x^{+}]={\mathbb{P}}_x[{\tau }_y<{\tau }_x^{+}].$$

$$\underset{n\to \infty }{\lim }{\mathbb{P}}_x^n[{\tau }_y<{\tau }_x^{+}]={\mathbb{P}}_x[{\tau }_y\le {\tau }_x^{+}].$$

$${\mathbb{P}}_B[{\tau }_T<{\tau }_B^{+}]$$

$$=1-{\mathbb{P}}_0[{\tau }_B\le {\tau }_T]$$

$$=1-{\mathbb{P}}_0[{\tau }_B<{\tau }_T]-{\mathbb{P}}_0[{\tau }_B={\tau }_T=\infty ].$$

$${\mathbb{P}}_B[{\tau }_T<{\tau }_B^{+}]={\mathbb{P}}_0[{\tau }_T<{\tau }_B]=\frac{1-{\mathbb{P}}_0[{\tau }_B={\tau }_T=\infty ]}{2}<\frac{1}{2}$$

$${\mathbb{P}}_B[{\tau }_T\le {\tau }_B^{+}]={\mathbb{P}}_0[{\tau }_T\le {\tau }_B]=\frac{1+{\mathbb{P}}_0[{\tau }_B={\tau }_T=\infty ]}{2}>\frac{1}{2}.$$

$${\mathbb{P}}_B[{\tau }_T<{\tau }_B^{+}]=\frac{2}{5}\text{ and }{\mathbb{P}}_B[{\tau }_T\le {\tau }_B^{+}]=\frac{3}{5}.$$

$$\frac{1}{c_B{\mathbb{P}}_B[{\tau }_T<{\tau }_B^{+}]}\ne R^F(B,T)$$

$$\frac{1}{c_B}{\mathbb{E}}_B\left[\sum _{k=0}^{{\tau }_T-1}\text{mathds}{1}_B({\omega }_k)\right]=\frac{1}{c_B{\mathbb{P}}_B[{\tau }_T\le {\tau }_B^{+}]}\ne R^F(B,T).$$

$${\mathbb{P}}_x(\gamma ):={\mathbb{P}}_x(\left\{\gamma \right\}\times V^{\mathbb{N}})=\text{mathds}{1}_x({\gamma }_0)\cdot \prod _{k=0}^{L(\gamma )-1}p({\gamma }_k,{\gamma }_{k+1}).$$

$${\Gamma }_G(x,y;A):=\left\{\gamma \in {\Gamma }_G(x,y)|{\gamma }_k\in A\text{ for all }k=0,\dots ,L(\gamma )\right\}$$

$$\underset{n\to \infty }{\lim }\sum _{\gamma \in {\Gamma }_G(x,y;V_n)}{\mathbb{P}}_x^n(\gamma )=\sum _{\gamma \in {\Gamma }_G(x,y)}{\mathbb{P}}_x(\gamma )$$

$${\partial }_{i}A:=\left\{v\in A|\exists w\in V\setminus A:c(v,w)>0\right\}$$

$$\overline{V_n}=V_n\cup \left\{\overline{v}|v\in {\partial }_i{V}_n\right\},$$

$$\overline{c_n}(x,y)=\overline{c_n}(y,x)=\{\begin{array}{ll}c(x,y)& ,x,y\in V_n\\ {\sum }_{z\notin V_n}c(x,z)& ,y=\overline{x}\\ 0& ,\text{ otherwise}\end{array}.$$

$$\overline{p_n}(x,y)=\frac{\overline{c_n}(x,y)}{(\overline{c_n}{)}_x}.$$

$$p_n(x,y)\ge p(x,y)=\overline{p_n}(x,y).$$

$$\underset{n\to \infty }{\lim }{p}_n(x,y)=p(x,y)=\overline{p_m}(x,y).$$

$${\Gamma }_{G_n}(x,y)={\Gamma }_{\overline{G_n}}(x,y;V_n)={\Gamma }_G(x,y;V_n).$$

Full text

Keywords. Weighted graph, Electrical networks, Effective resistance, Random walk, Transience

AMS Subject Classification (2010): Primary 05C63 $\cdot$ 05C81; Secondary 60J10 $\cdot$ 05C12

1 Introduction

We consider undirected, connected graphs with no multiple edges and no self-loops. Each edge $(x,y)$ is given a positive weight $c(x,y)$. A possible interpretation is that $(x,y)$ is a resistor with resistance $1/c(x,y)$. The graph then becomes an electrical network.

More precisely, a graph $G=(V,c)$ consists of an at most countable set of vertices $V$ and a weight function $c:V\times V\to\mathbb{R}_{\geq 0}$ such that $c$ is symmetric and for all $x\in V$, we have $c(x,x)=0$ and

[TABLE]

We think of two vertices $x,y$ as being adjacent if $c(x,y)>0$.

For $x\in V$, let $\mathbb{P}_{x}$ be the random walk on $G$ starting in $x$. It is the Markov chain defined by the transition matrix

[TABLE]

and initial distribution $\delta_{x}$. We will think of $\mathbb{P}_{x}$ as a probability measure on $\Omega=V^{\mathbb{N}_{0}}$ equipped with the $\sigma$-algebra $(2^{V})^{\otimes\mathbb{N}_{0}}$. If not explicitly stated otherwise, we will from now on assume that every occurring graph is connected. In that case, $\mathbb{P}_{x}$ is irreducible.

For a set of vertices $A\subseteq V$, let

[TABLE]

be hitting times of $A$. For $x\in V$, we use the shorthand notation $\tau_{\left\{x\right\}}=:\tau_{x}$.

Suppose that $G$ is finite. Ohm’s Law states that the effective resistance $R(x,y)$ between to vertices $x,y$ is the voltage drop needed to induce an electrical current of exactly 1 ampere from $x$ to $y$.

The relationship between electrical currents and the random walk of $G$ has been studied intensively [DS84, JP09, LP16, Tet91]. For finite graphs, $x\neq y$, one has the following probabilistic representations

[TABLE]

Note that $(c_{z})_{z\in V}$ is an invariant measure of $p$. A proof of the first equality in the unweighted case can be found in [Tet91] and can be extended to fit our more general context. To see that (1.1) equals (1.2), realize that $\sum_{k=0}^{\tau_{y}-1}\mathds{1}_{x}$ is geometrically distributed with parameter $\mathbb{P}_{x}[\tau_{x}^{+}<\tau_{y}]$. For the last equality, use that any finite graph is recurrent and thus $\mathbb{P}_{x}[\tau_{x}^{+}=\tau_{y}=\infty]=0$.

The subject of effective resistances gets much more complicated on infinite graphs since those may admit multiple different notions of effective resistances. Recurrent graphs, however, have a property which is often referred to as unique currents [LP16] and consequently also have one unique effective resistance. In this case, the above representation holds [Bar17, Wei18]. Indeed, [Bar17, Theorem 2.61] states the more general inequalities

[TABLE]

for the free effective resistance $R^{F}$ (see Section 2) of any infinite graph.

In [JP09, Corollary 3.13 and 3.15], it is suggested that one has

[TABLE]

on all transient networks. However, this is false as our example in Section 3 shows.

The main result of this work (Corollary 6.3) states that the free effective resistance of a transient graph $G=(V,c)$ admits the representation (1.5) for all $x,y\in V$ if and only if $G$ is a subgraph of an infinite line. Corollary 6.5 states that the lower bound in (1.4) are attained if and only if $G$ is recurrent.

2 Free effective resistance

Let $G=(V,c)$ be an infinite connected graph. For any $W\subseteq V$, let $G\mathord{\restriction}_{W}:=(W,c\mathord{\restriction}_{W})$ be the subgraph of $G$ induced by $W$. We say a sequence $(V_{n})_{n\in\mathbb{N}}$ of subsets of $V$ is a finite exhaustion of $V$ if $|V_{n}|<\infty$, $V_{n}\subseteq V_{n+1}$ and $V=\cup_{n\in\mathbb{N}}V_{n}$. Define $G_{n}=(V_{n},c_{n}):=G\mathord{\restriction}_{V_{n}}$.

Definition 2.1**.**

Let $(V_{n})_{n\in\mathbb{N}}$ be any finite exhaustion of $V$ such that $G_{n}$ is connected. For $x,y\in V$, the free effective resistance $R^{F}(x,y)$ of $G$ is defined by

[TABLE]

Remark 2.2**.**

The fact that $R_{G_{n}}(x,y)$ converges is due to Rayleigh’s monotonicity principle (see e.g. [BLPS01, Gri10]).

We denote by $\mathbb{P}^{n}_{x}$ the random walk on $G_{n}$ starting in $x$ with transition matrix $p_{n}$. Since we can extend it to a function on $V$ by defining $p_{n}(x,y)=0$ whenever $x\notin V_{n}$ or $y\notin V_{n}$, $\mathbb{P}^{n}_{x}$ is a probability measure on $\Omega=V^{\mathbb{N}_{0}}$ and we have

[TABLE]

for all $x,y\in V_{n}$.

Remark 2.3**.**

Note that

[TABLE]

Since

[TABLE]

for all $n\in\mathbb{N}$ and $c_{x}=\lim_{n\to\infty}(c_{n})_{x}$, (1.5) holds if and only if

[TABLE]

Analogously, the lower bound of (1.4) is attained if and only if

[TABLE]

3 The transient $\mathcal{T}$

We will now show that (1.5) does not hold in general. Consider the graph $\mathcal{T}$ shown in Figure 1. It is transient and we have $R^{F}(B,T)=2$. However,

[TABLE]

Due to the symmetry of $\mathcal{T}$ we have $\mathbb{P}_{0}[\tau_{B}<\tau_{T}]=\mathbb{P}_{0}[\tau_{T}<\tau_{B}]$. Together with the transience of $\mathcal{T}$, this implies

[TABLE]

and

[TABLE]

More precisely, one can compute

[TABLE]

Hence,

[TABLE]

and

[TABLE]

Remark 3.1**.**

Note that, although $\mathcal{T}$ is transient, it has unique currents since every harmonic function is constant. This shows that whether (1.5) holds is more tightly connected to the transience of the graph than to the uniqueness of currents (which is equivalent to the existence of harmonic Dirichlet functions [LP16]).

4 Probability of paths

To check whether (2.1) holds, it is useful to write both sides as sums of probabilities of paths.

A sequence $\gamma=(\gamma_{0},\ldots,\gamma_{n})\in V^{n+1}$ is called a path (of length n) in $G$ if $c(\gamma_{k},\gamma_{k+1})>0$ for all $k=0,\ldots,n-1$. We denote by $L(\gamma)$ the length of $\gamma$ and by $\Gamma_{G}$ the set of all paths in $G$. A path $\gamma$ is called simple if it does not contain any vertex twice. The probability of $\gamma$ with respect to $\mathbb{P}_{x}$ is defined by

[TABLE]

We say $\gamma$ is $x\to y$ if $\gamma_{0}=x,\gamma_{L(\gamma)}=y$ and $\gamma_{k}\notin\{x,y\}$ for all $k=1,\ldots,L(\gamma)-1$. We denote by $\Gamma_{G}(x,y)$ the set of all paths $x\to y$ in $G$.

For $A\subseteq V$, let

[TABLE]

be the set of all paths $x\to y$ in $G$ which only use vertices in $A$.

Using this notion and $\Gamma_{G_{n}}(x,y)=\Gamma_{G}(x,y;V_{n})$, (2.1) becomes

[TABLE]

Since $\Gamma_{G}(x,y;V_{n})$ increases to $\Gamma_{G}(x,y)$, this looks like an easy application of either the Monotone Convergence Theorem or the Dominated Convergence Theorem. However, both are not applicable since $\mathbb{P}^{n}_{x}(\gamma)$ may be strictly greater than $\mathbb{P}_{x}(\gamma)$.

To investigate when exactly (4.1) holds, we will introduce another random walk on $V$ which can be considered an intermediary between $\mathbb{P}^{n}_{x}$ and $\mathbb{P}_{x}$.

5 Extended finite random walk

The only difference in the behavior of $\mathbb{P}_{x}$ and $\mathbb{P}^{n}_{x}$ occurs when $\mathbb{P}_{x}$ leaves $V_{n}$. Instead, $\mathbb{P}^{n}_{x}$ is basically reflected back to a vertex in $V_{n}$. We will now construct an intermediary random walk which still has a finite state space, models the behavior of stepping out of $V_{n}$ and has the same transition probabilities as $\mathbb{P}_{x}$ in $V_{n}$. This is done by adding boundary vertices to $G_{n}$ wherever there is an edge from $V_{n}$ to $V\setminus V_{n}$.

For any set $A\subseteq V$, let

[TABLE]

be the inner boundary and $\partial_{o}A:=\partial_{i}(V\setminus A)$ be the outer boundary of $A$ in $G$.

For any $v\in\partial_{i}A$, let $\overline{v}$ be a copy of $v$. Define $\overline{G_{n}}=(\overline{V_{n}},\overline{c_{n}})$ where

[TABLE]

and $\overline{c_{n}}$ is defined as follows. For $x,y\in\overline{V_{n}}$, let

[TABLE]

In particular, we have $(\overline{c_{n}})_{x}=c_{x}$ for all $x\in V$. We denote by $\overline{\mathbb{P}^{n}_{x}}$ the random walk on $\overline{G_{n}}$ starting in $x$ with transition matrix $\overline{p_{n}}$ given by

[TABLE]

Furthermore, let $V_{n}^{*}:=\overline{V_{n}}\setminus V_{n}$.

Example 5.1**.**

Let $G$ be the lattice $\mathbb{Z}^{2}$ with unit weights, see Figure 2. Furthermore, let $V_{n}:=\left\{-n\ldots,0,\ldots,n\right\}^{2}$.

$G_{1}$ and $\overline{G_{1}}$ can be seen in Figure 3. Note that $c((1,1),\overline{(1,1)})=2$ since $(1,1)$ has two edges leaving $V_{1}$ in $G$.

Lemma 5.2** (Relation of $p_{n},\overline{p_{n}}$ and $p$).**

For $x,y\in V_{n}$ we have

[TABLE]

For $x,y\in V$ and $m\in\mathbb{N}$ such that $x,y\in V_{m}$, we have

[TABLE]

Note that for $n\in\mathbb{N}$, we have

[TABLE]

By Lemma 5.2, the following holds for all $x,y\in V_{n}$.

[TABLE]

The connection between $\mathbb{P}_{x}^{n}(\gamma)$ and $\overline{\mathbb{P}^{n}_{x}}(\gamma)$ is a bit more intricate.

Definition 5.3**.**

For $x,y\in V_{n}$, let $\pi:\Gamma_{\overline{G_{n}}}(x,y)\to\Gamma_{G_{n}}(x,y)$ be the projection of $\Gamma_{\overline{G_{n}}}(x,y)$ onto $\Gamma_{G_{n}}(x,y)$ which removes all steps of the form $(\overline{v},v)$.

More precisely, for $\gamma\in\Gamma_{G_{n}}(x,y)$, let $\pi(\gamma)=\gamma$. For general $\gamma\in\Gamma_{\overline{G_{n}}}(x,y)$, define $\pi(\gamma)$ inductively over $L(\gamma)$. If $L(\gamma)\leq 2$, then $\gamma\in\Gamma_{G_{n}}(x,y)$ and thus $\pi(\gamma)=\gamma$. For $n:=L(\gamma)>2$, let $\gamma=(x,\gamma_{1},\ldots,\gamma_{n-1},y)$ and $\gamma^{\prime}=(x,\gamma_{1},\ldots,\gamma_{n-3})$. Now, $(\gamma_{n-2},\gamma_{n-1},y)$ can either be of the form $(v,w,y)$ with $v,w\in V_{n}$ or $(\overline{v},v,y)$ with $v\in V_{n}$. In the former case, let

[TABLE]

in the latter

[TABLE]

Lemma 5.4**.**

For all $x\neq y$ and $\gamma\in\Gamma_{G_{n}}(x,y)$, we have

[TABLE]

Proof.

We prove the claim via induction over $L(\gamma)$. For $L(\gamma)=1$, we have $\gamma=(x,y)$. By definition of $\pi$, any preimage $\gamma^{\prime}\in\pi^{-1}(\gamma)$ is in $\Gamma_{\overline{G_{n}}}(x,y)$ and thus visits $x$ and $y$ exactly once. Hence, $\pi^{-1}(\gamma)=\left\{\gamma\right\}$ and

[TABLE]

Suppose that $n:=L(\gamma)>1$ and let $\gamma^{\prime}=(\gamma_{0},\ldots,\gamma_{n-1})$ and $v=\gamma_{n-1}$ Then,

[TABLE]

and

[TABLE]

Since

[TABLE]

we have

[TABLE]

Hence,

[TABLE]

∎

Proposition 5.5**.**

For $x,y\in V_{n}$, we have

[TABLE]

Proof.

Using

[TABLE]

we compute

[TABLE]

∎

Since we now have clarified the relation between $\mathbb{P}^{n}_{x}$, $\overline{\mathbb{P}^{n}_{x}}$ and $\mathbb{P}_{x}$, we can return our attention to (2.1).

Proposition 5.6**.**

For $x,y\in V$, $x\neq y$, we have

[TABLE]

if and only if

[TABLE]

Proof.

We have

[TABLE]

and

[TABLE]

Since $\Gamma_{G}(x,y;V_{n})=\Gamma_{\overline{G_{n}}}(x,y;V_{n})$ and $(c_{n})_{x}\to c_{x}$, it follows that $\mathbb{P}^{n}_{x}[\tau_{y}<\tau_{x}]\to\mathbb{P}_{x}[\tau_{y}<\tau_{x}]$ holds if and only if

[TABLE]

This is the same as

[TABLE]

∎

Using the same approach, we can also characterize when (2.2) holds.

Proposition 5.7**.**

For $x,y\in V$, $x\neq y$, we have

[TABLE]

if and only if

[TABLE]

which in turn is equivalent to

[TABLE]

Proof.

Using (5.2) and (5.3) from the proof of Proposition 5.6, we have

[TABLE]

and

[TABLE]

Hence, we have convergence as desired if and only if

[TABLE]

On the other hand, we have

[TABLE]

which implies the second claim. ∎

Remark 5.8**.**

An equivalent approach would be to consider a lazy random walk on $G_{n}$ which has the same transition probabilities $p(v,w)$ as $\mathbb{P}_{x}$ for $v\neq w$ but stays at $v$ with probability

[TABLE]

In that case the notion of ”stepping out of $V_{n}$” would be modeled by staying at any vertex $v\in V_{n}$.

6 Embedding $\mathcal{T}$ into transient graphs

We will show that whenever a graph $G$ is transient and not part of an infinite line, one can find a subgraph of $G$ which is similar to $\mathcal{T}$ from Section 3. We will also show that this is sufficient for (5.1) not to hold.

Proposition 6.1**.**

Let $G$ be a transient, connected graph which is not a subgraph of a line. Then, there exist $x,y,z\in V$ such that $x\neq y$, $(x,z,y)$ is a path in $G$ and

[TABLE]

Proof.

Since $G$ is transient, it is infinite. If $G$ is not a subgraph of a line, then there exists some $z\in V$ with at least three adjacent vertices. Let $F$ be a set of exactly three neighbors of $z$. Since $G$ is transient and $F$ is finite, there exists $v\in\partial_{o}F$ such that

[TABLE]

If $v=z$, we can choose $x,y\in F$, $x\neq y$, and get

[TABLE]

If $v\neq z$, then there exists $v^{\prime}\in F$ such that $(z,v^{\prime},v)$ is a path in $G$. Let $x,y\in V$ such that $F=\left\{x,y,v^{\prime}\right\}$, see Figure 4. It follows that

[TABLE]

∎

Theorem 6.2**.**

Let $G$ be a transient, connected graph. Then,

[TABLE]

holds if and only if $G$ is a subgraph of an infinite line.

Proof.

First, assume that $G$ is a subgraph of an infinite line and let $x,y\in V$, $x\neq y$. Then, for any $n\in\mathbb{N}$ sufficiently big, we have

[TABLE]

i.e. there exists no path $x\to y$ which leaves $V_{n}$ before reaching $y$. Hence,

[TABLE]

To prove the converse direction, suppose that $G$ is not a subgraph of a line. By Proposition 6.1, we know that there exist distinct vertices $x,y,z\in V$ such that $(x,z,y)$ is a path in $G$ and $\mathbb{P}_{z}[\tau_{x}=\tau_{y}=\infty]>0$. Hence,

[TABLE]

Without loss of generality assume that $\limsup_{n\to\infty}\overline{\mathbb{P}_{z}^{n}}[\tau_{V_{n}^{*}}<\tau_{y}<\tau_{x}]>0$. It follows that $\limsup_{n\to\infty}\overline{\mathbb{P}_{x}^{n}}[\tau_{V_{n}^{*}}<\tau_{y}<\tau_{x}^{+}]>0$ because for all $n\in\mathbb{N}$, we have

[TABLE]

∎

Corollary 6.3**.**

Let $G$ be a transient, connected graph. Then,

[TABLE]

holds for all $x,y\in V$ if and only if $G$ is a subgraph of an infinite line.

Proof.

As seen in (2.1), the desired probabilistic representation (1.5) holds if and only if

[TABLE]

By Proposition 5.6, this is equivalent to

[TABLE]

and the claim follows by Theorem 6.2. ∎

Theorem 6.4**.**

Let $G$ be an infinite graph. If

[TABLE]

holds, then $G$ is recurrent.

Proof.

By Proposition 5.7, we have

[TABLE]

Suppose that $G$ is transient and not a subgraph of a line. Using the same arguments as in the proof of Theorem 6.2, we see that there exist distinct vertices $x,y,z\in V$ such that $(x,z,y)\in\Gamma_{G}(x,y)$ and

[TABLE]

Since

[TABLE]

it follows that

[TABLE]

which implies

[TABLE]

However, we also have

[TABLE]

and it follows that

[TABLE]

which is a contradiction to (6.1).

Hence, if $G$ is transient, then it must be a subgraph of a line. In this case,

[TABLE]

follows for all $x,y\in V$ by Theorem 6.2. Together with (6.1), this implies

[TABLE]

for all $x,y\in V$. However, this is a contradiction to the transience of $G$.

∎

Corollary 6.5**.**

Let $G$ be an infinite, connected graph. Then,

[TABLE]

holds for all $x,y\in V$ if and only if $G$ is recurrent.

Proof.

If $G$ is recurrent, we have $\mathbb{P}_{x}[\tau_{x}^{+}=\tau_{y}=\infty]=0$ for all $x,y\in V$. Hence,

[TABLE]

and (1.4) implies the claim.

If (6.2) holds for all $x,y\in V$, then we have

[TABLE]

for all $x,y\in V$ and Theorem 6.4 implies the recurrence of $G$. ∎

This shows that the lower bound in (1.4) is actually a strict inequality for transient graphs.