Let \(J\) be the all-ones matrix, and let \(A\) denote the adjacency matrix of a graph. It is known that if two graphs are cospectral with respect to \(yJ-A\) for two distinct values of \(y\), then they are cospectral for all \(y\). In this paper the authors focus on \(\hat y\)-cospectral graphs, which are graphs cospectral with respect to \(yJ-A\) for exactly one value \(\hat y\) of \(y\). They prove that such a \(\hat y\) must always be a rational number and prove existence of a pair of \(\hat y\)-cospectral graphs for every rational \(\hat y\). A computer construction has been used to generate all \(\hat y\)-cospectral graph pairs on at most nine vertices and also all pairs of \(\hat y\)-cospectral graph pairs on at most eleven vertices for all rational \(\hat y\in(0,1)\), where one graph is regular and the other one is not.