A Note on Metric Properties of Infinite Graphs

Abstract

Throughout this note by a graph G we mean an undirected connected graph without loops or multiple edges, defined on a countably infinite vertex set V(G). The edge set of G is denoted by E(G) and the valency of a vertex x of G by v(x:G). A graph G is said to have universally bounded valencies if there is positive number L such that v(x:G) = L for all x ∈ V(G).