Isoperimetric numbers of graphs

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For X⊆ V( G), let ∂X denote the set of edges of the graph G having one end in X and the other end in V( G) βX. The quantity i(G)≔ min{ |∂X| |X| } , where the minimum is taken over all non-empty subsets X of V( G) with |X| ≤ |V(G)| 2 , is called the isoperimetric number of G. The basic properties of i( G) are discussed. Some upper and lower bounds on i( G) are derived, one in terms of | V( G)| and | E( G)| and two depending on the second smallest eigenvalue of the difference Laplacian matrix of G. The upper bound is a strong discrete version of the wellknown Cheeger inequality bounding the first eigenvalue of a Riemannian manifold. The growth and the diameter of a graph G are related to i( G). The isoperimetric number of Cartesian products of graphs is studied. Finally, regular graphs of fixed degree with large isoperimetric number are considered.