Strongly subharmonic functions,graphs, and their asymptotic growth

Let G be an infinite connected graph with uniformly bounded vertex degree, and let denote the Laplace operator corresponding to the simple random walk on it. In this paper we obtain relations between the structure of the graph and the qualitative behaviour of the class of functions u satisfying ub>0, namely we relate the asymptotic growth of the function u and that of the cardinality of balls in G.Acknowledgement We wish to thank the referee for her/his precise comments and helpful suggestions.