On the spectral radius of bipartite graphs which are nearly complete

Abstract

For p, q, r, s, t ? Z+ with rt ? p and st ? q, let G = G(p, q; r, s; t) be the bipartite graph with partite sets U = {u1, . . . , up} and V = {v1, . . . , vq} such that any two edges ui and vj are not adjacent if and only if there exists a positive integer k with 1 ? k ? t such that (k - 1)r + 1 ? i ? kr and (k - 1)s + 1 ? j ? ks. Under these circumstances, Chen et al. (Linear Algebra Appl. 432:606-614, 2010) presented the following conjecture: Assume that p ? q, k < p, |U| = p, |V| = q and |E(G)| = pq - k. Then whether it is true that ?1(G) ? ?1(G(p, q; k, 1; 1)) = - pq - k + - p2q2 - 6pqk + 4pk + 4qk2 - 3k2 2 . In this paper, we prove this conjecture for the range minvh?V {deg vh} ? -p-1 2 -. © 2013 Das et al.