Cospectrality of graphs

Richard Brualdi proposed in Stevanivic (2007) [6] 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 study Problem A and determine the cospectrality of certain graphs by the Euclidian distance. Let K-n denote the complete graph on n vertices, nK(1) denote the null graph on n vertices and K-2 + (n - 2)K-1 denote the disjoint union of the K-2 with n - 2 isolated vertices, where n >= 2. In this paper we find cs(K-n), cs(nK(1)), cs(K-2 + (n - 2)K-1) (n >= 2) and cs(K-n,K-n). (C) 2014 Elsevier Inc. All rights reserved.