Isoperimetric numbers of graphs

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Abstract

For XV(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.

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This work was done while the author was visiting the Simon Fraser University, Burnaby, BC, Canada.