A cospectral construction for the generalized distance matrix

Abstract The generalized distance matrix of a graph is a matrix in which the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>,</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(i,j) th entry is a function, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> </m:math> f , of the distance between vertex <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>i</m:mi> </m:math> i and vertex <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>j</m:mi> </m:math> j . Depending on the choice of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> </m:math> f , 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q raised to the power of the distance between the vertices. We give an upper bound on the values of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q needed to show a pair of graphs is cospectral for all values of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q corresponding to the diameter of the graphs. We also give cospectral constructions unique to value <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>⁄</m:mo> <m:mn>2</m:mn> </m:math> q=1/2 .