On the spectral radius of bipartite graphs which are nearly complete
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Date
2013-12
Authors
Das, Kinkar Chandra
Maden, Ayşe Dilek
Çevik, Ahmet Sinan
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
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.
Description
Keywords
Mathematics, Bipartite graph, Adjacency matrix, Spectral radius, Eigenvalues, Conjectures, Bounds, Proof
Citation
Das, K. C. vd. (2013). “On the spectral radius of bipartite graphs which are nearly complete”. Journal of Inequalities and Applications, 2013.