Abstract
Two classes of infinite graphs with unbounded vertex degrees are introduced and studied. These are g-tempered and strongly g-tempered graphs, respectively. In such graphs, the degree growth is controlled by a function g:R+→R+. It is proven that for g(t)=logt (resp. g(t)=tlogt), the number of simple paths of length N originated at a given vertex x (resp. the number of finite connected subgraphs of order N containing x) is exponentially bounded in N for N belonging to an infinite subset Nx⊂N, which is a sequence {Nk} for g-tempered (resp. {N:N≥Nx} for strongly g-tempered) graphs. It is shown that the graphs in which the path distance between vertices of large degree is bigger than a certain function of their degrees belong to the classes introduced in this paper. These results are then applied to a number of problems, including estimating volume growth, the growth of the Randić index, and of the number of greedy animals.
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