A cospectral construction for the generalized distance matrix

The generalized distance matrix of a graph is a matrix in which the (i,j)\left(i,j)th entry is a function, ff, of the distance between vertex ii and vertex jj. Depending on the choice of ff, this family of matrices includes both the adjacency matrix and the traditional distance matrix. We present a cospectral construction for the generalized distance matrix akin to Godsil-McKay Switching. We also investigate a special case of the generalized distance matrix: the exponential distance matrix, which is a matrix where every entry is a value qq raised to the power of the distance between the vertices. We give an upper bound on the values of qq needed to show a pair of graphs is cospectral for all values of qq corresponding to the diameter of the graphs. We also give cospectral constructions unique to value q=1⁄2q=1/2.

Abstract licence: CC BY

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