A Diagonalization Algorithm for the Distance Matrix of Cographs

Cographs is a well-known class of graphs in graph theory, which can be generated from a single vertex by applying a series of complement (or equivalently join operations) and disjoint union operations. The distance spectrum of graphs is a rather active topic in spectral graph theory these years. This paper denotes to revealing some properties for the distance spectrum of cographs. More precisely, we present an algorithm, using
$O(n)$
time and space, to diagonalize the distance matrix of cographs, from which one can deduce a diagonal matrix congruent to matrix
$D + \lambda I$
, where
$D$
is the distance matrix of a cograph,
$\lambda $
is a real number, and
$I$
is the identity matrix. Besides, we also give some applications of such algorithm about the inertia of distance matrix of complete multipartite graphs.