The Cheeger constant, the heat kernel, and the Green kernel of an infinite graph (Q1404226)

An infinite graph \({\mathcal G}= (V, E)\) consists of a set \(V\) of an infinite number of vertices and a set \(E\) of edges connecting two vertices. \({\mathcal G}\) is connected and locally finite if for each \(x\in V\), the degree \(m(x)= \#\{y\in V\mid y\sim x\}\) is finite and \(>0\), where \(y\sim x\) means that \(x\) and \(y\) are connected by an edge. The graph distance \(d(x, y)\) for \(x,y\in V\) is the minimum length of a path connecting \(x\) and \(y\). Let \(u_t(x, y)\) be the \(t\)-step transition probability (called also heat kernel) from \(x\) to \(y\) for \(t\geq 1\). The Green kernel is defined as \(G(x, y)= \sum_t u_t(x,y)\). The main purpose of this paper is to give lower and upper estimations of the heat kernel \(u_t(x, y)\) and the Green kernel \(G(x, y)\) in terms of the degrees and the distance \(d(x,x_0)\) for an infinite graph \({\mathcal G}\), which is assumed to be connected and locally finite.