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 by of y. We call such graphs by-cospectral. It follows that by is a rational number, and we prove existence of a pair of by-cospectral graphs for every rational by. In addition, we generate by computer all by-cospectral pairs on most nine vertices. Recently, Chesnokov and the second author constructed pairs of by-cospectral graphs for all rational by 2 (0; 1), where one graph is regular and the other one is not. This phenomenon is only possible for the mentioned values of by, and by computer we -nd all such pairs of by-cospectral graphs on at most eleven vertices.
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Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number
31.