Distance between spectra of graphs

Richard Brualdi proposed in Stevanivic (2007) [10] the following problem: (Problem AWGS.4) Let G(n) and G'(n) be two nonisomorphic graphs on n vertices with spectra lambda(1) >= lambda(2) >= ... >= lambda(n) and lambda'(1) >= lambda'(2) >= ... >= lambda'(n), respectively. Define the distance between the spectra of G(n) and G'(n) as lambda(G(n), G'(n)) = Sigma(n)(i=1)(lambda(i) - lambda'(i))(2) (or use Sigma(n)(i=1) vertical bar lambda(i) - lambda'(i)vertical bar). Define the cospectrality of G(n) by cs(G(n)) = min{lambda(G(n), G'(n)) : G'(n) not isomorphic to G(n)}. Let cs(n) = max{cs(G(n)) : G(n) a graph on n vertices}. Problem A. Investigate cs(G(n),) for special classes of graphs. Problem B. Find a good upper bound on cs(n). In this paper we completely answer Problem B by proving that cs(n) = 2 for all n >= 2, whenever cs(n) is computed with respect to any l(p)-norm with 1 <= p <= infinity and cs(n) = 1 with respect to the l(infinity)-norm. The cospectrality cs(K-m,K-n) of the complete bipartite graph K-m,K-n for all positive integers m and n with m <= n < m - 1 + 2 root m-1 is computed. As a consequence of the latter result, it is proved that in general there is no constant upper bound on vertical bar e - e'vertical bar, where e and e' denote the number of edges of the graphs G and G', respectively, such that cs(G) = lambda(G, G'). In contrast we show that for such graphs G and G', we have vertical bar root e - root e'vertical bar <= 1. In particular, vertical bar e - e'vertical bar <= 3e. (C) 2014 Elsevier Inc. All rights reserved.