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Let J be the all-ones matrix, and let A denote the adjacency matrix of a graph. An old result of Johnson and Newman states that if two graphs are cospectral with respect to yJ - A for two distinct values of y, then they are cospectral for all y. Here we will focus on graphs cospectral with respect to yJ - A for exactly one value widehat{y} of y. We call such graphs widehat{y}-cospectral. It follows that widehat{y} is a rational number, and we prove existence of a pair of widehat{y}-cospectral graphs for every rational widehat{y}. In addition, we generate by computer all widehat{y}-cospectral pairs on most nine vertices. Recently, Chesnokov and the second author constructed pairs of widehat{y}-cospectral graphs for all rational widehat{y}{in}(0,1), where one graph is regular and the other one is not. This phenomenon is only possible for the mentioned values of widehat{y}, and by computer we find all such pairs of widehat{y}-cospectral graphs on at most eleven vertices.

Cospectral Graphs and the Generalized Adjacency Matrix