On the largest eigenvalues of bipartite graphs which are nearly complete

For positive integers p, q, r, s and t satisfying rt ≤ p and st ≤ q, let G (p, q ; r, s ; t) be the bipartite graph with partite sets {u₁, ..., u_p} and {v₁, ..., v_q} such that 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. In this paper we study the largest eigenvalues of bipartite graphs which are nearly complete. We first compute the largest eigenvalue (and all other eigenvalues) of G (p, q ; r, s ; t), and then list nearly complete bipartite graphs according to the magnitudes of their largest eigenvalues. These results give an affirmative answer to [1, Conjecture 1.2] when the number of edges of a bipartite graph with partite sets U and V, having | U | = p and | V | = q for p ≤ q, is pq - 2. Furthermore, we refine [1, Conjecture 1.2] for the case when the number of edges is at least pq - p + 1.