A New Upper Bound on the Cheeger Number of a Graph

where X(G) is the set of all parts X/V(G) so that X{< and X =V(G)"X{<. Also, E(A, B) is the set of all A B edges in G and Vol(A)= x # A mG(x). Here mG(x) is the degree of the vertex x in G. For instance, the Cheeger number of the complete graph K on N vertices is h(K)=N [2(N&1)]. Also, the Cheeger number of a claw (or star) K1, N=K V K is h(K1, N)=1. In general, h(G) 1. Given n # Z, n>0, let G(n) be the family of all finite connected graphs G admitting at least two edges e, e$ # E(G) with dG(e, e$) 2n+2. Here dG(e, e$) is the distance between the edges e and e$, i.e., the minimum number of edges in a path that connects a vertex of e and a vertex of e$. For instance, for any path P on 2m+1 vertices (m 3), P # G(n) for any 1 n m&2. Yet, K G(n) and K1, N G(n), for any n>0. Let $(G) and 2(G) be the minimum and maximum degrees of a vertex in G, respectively. We may state