Expansion Constants and Hyperbolic Embeddings of Finite Graphs

Abstract

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.