On the spectral radius of bipartite graphs which are nearly complete

Abstract

For p, q, r, s, t is an element of Z(+) with rt <= p and st <= q, let G = G(p, q; r, s; t) be the bipartite graph with partite sets U = {u(1), ..., u(p)} and V = {v(1),..., v(q)} such that any two edges u(i) and v(j) 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, vertical bar U vertical bar = p, vertical bar V vertical bar = q and vertical bar E(G)vertical bar = pq - k. Then whether it is true

that lambda(1)(G) <= lambda(1)(G(p, q; k, 1; 1)) = root pq - k + root p(2)q(2) - 6pqk + 4pk + 4qk(2) - 3k(2)/2.

In this paper, we prove this conjecture for the range min(vh is an element of V){deg v(h)} <= left perpendicular p-1/2right perpendicular.