Structure and Density of Sparse Crossbar

A sparse crossbar (n,m)-concentrator is a bipartite graph with n source and m sink vertices, m ≤ n, in which there exists a matching between every m source vertices and the m sink vertices. In this paper, we investigate the structure, and the density of sparse crossbar (n,m)-concentrators among all 2 bipartite graphs. We establish that the density of sparse crossbar concentrators is bounded from below by 0.2887 when m = n, from above by 1/e when m = n/2, and it tends to 0 when m = 1, as n → ∞. We also derive upper and lower bounds on the density of sparse crossbar (n,m)-concentrators for an arbitrary m ≤ n. The lower bounds provide an insight into the structure of sparse crossbar concentrators, while the upper bounds give a partial characterization of bipartite graphs which fail to have a concentrator property. This work is supported in part by the National Science Foundation under grant No. NCR9405539.