Two conjectures on the cospectrality of complete tripartite graphs are false

Source abstract

For a graph G of order n with adjacency eigenvalues lambda_1(G) >= ... >= lambda_n(G), the cospectrality of G within the set G of all graphs of order n is the smallest spectral distance from G to another graph of G. Abdollahi and Zakeri conjectured that for every n >= 4 the cospectrality of the complete tripartite graph K_{n,n,n}, both in the Manhattan and in the Euclidean spectral distance, is attained precisely at K_{n-1,n,n+1}. We show that both conjectures are false for every n >= 4, that is, in the whole range in which they are stated. The counterexample is the graph H_n obtained from K_{n-1,n,n} by adding a vertex adjacent to the part of cardinality n - 1, and it is the natural generalization of the graph which the authors themselves exhibited for n = 3 and considered to be exceptional. We determine the spectrum of H_n in a closed form, and we show that the two spectral distances tend to 1 and to 1/2 respectively, instead of to (2*sqrt(3))/3 and to 2/3 as conjectured. On the other hand, we show that in the Euclidean distance the conjecture becomes asymptotically true when the competing graphs are restricted to complete multipartite graphs, so that it fails because a graph of rank four beats the conjectured minimizer.