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In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

In this paper, we study a finite connected graph which admits a quasimonomorphism to hyperbolic spaces and give a geometric bound for the Cheeger constants in terms of the volume, an upper bound of the degree, and the quasimonomorphism. §1. Introduction. We consider a graph G = (V, E) with the set of vertices V and the set of edges E that consists of pairs of vertices. In this paper, a graph admits no loops and multiple edges, and the set of vertices is finite or countably infinite. A graph is considered as a metric space with its graph distance d G .There are various results known on geometric bounds for the Cheeger constant, or the expansion constant h(G) and the second eigenvalue, or the first positive eigenvalue λ 2 (G) of the combinatorial Laplacian of a connected finite graph G = (V, E), in terms of an upper bound of the degree, d, the cardinality of V , |V |, the diameter of G, diam(G), and other characteristic properties.

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