The alternating polynomials and their relation with the spectra and conditional diameters of graphs (Q1356468)

If \(G\) is a graph on \(n\) vertices, then the distance \(\partial(G_1,G_2)\) between two subgraphs \(G_1,G_2\subset G\) is the minimum among the distances between vertices of \(G_1\) and \(G_2\). In this paper some bounds of the conditional \((s,t)\)-diameter \((1\leq s,t\leq n)\) defined by \[ D_{(s,t)}= \max_{G_1,G_2\subset G}\{\partial(G_1,G_2):|V(G_1)|= 1\text{ and }|V(G_2)|=t\}, \] in terms of the eigenvalues of the graph, are obtained. By using the Laplacian spectrum, similar results for locally regular digraphs are also given.