Strongly subharmonic functions, graphs, and their asymptotic growth

Let G be an infinite connected graph with uniformly bounded vertex degree, and let Delta 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 Deltau greater than or equal to b > 0, namely we relate the asymptotic growth of the function u and that of the cardinality of balls in G.