Graphs that are cospectral for the distance Laplacian

Abstract

The distance matrix D(G) of a graph G is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is D L (G) = T (G) − D(G), where T (G) is the diagonal matrix of row sums of D(G). We establish several general methods for producing D L -cospectral graphs that can be used to construct infinite families. We provide examples showing that various properties are not preserved by D L -cospectrality, including examples of D L -cospectral strongly regular and circulant graphs. We establish that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., |δ L

Constructions in combinatorics via neural networks

“…This question has a long history, for an overview on the extensive research on this topic we refer the reader to the recent survey [31] and the references therein, in particular [14]. Hogben-Reinhart [31] put a lot of emphasis on the spectral properties of transmission regular graphs -indeed, as shown in Table 7.2 in their survey, it is the last of the natural graph properties considered in [31] for which is not known whether it is preserved by D L -cospectrality.…”

Cospectral constructions for several graph matrices using cousin vertices

“…The cospectral construction given in [2] which we wish to extend starts with a set of vertices with special proprieties called cousins.…”

“…Cospectral constructions have been studied for the adjacency matrix [6], combinatorial Laplacian [5,7], signless Laplacian [7], normalized Laplacian [3], distance matrix [8], and to a lesser extent distance Laplacian matrix [1,2]. The adjacency matrix A has a 1 in the i, j entry if there is an edge between vertices i, j and 0 otherwise.…”

“…Additionally, we will generalize this construction and extend it to the signless Laplacian in Section 2. This generalization is based on a cospectral construction for the distance Laplacian given in [2] and an example is shown in Figure 3.…”

“…Despite there being no known cospectral trees for the distance Laplacian, there are numerous examples of cospectral graphs [1]. A cospectral construction method given in [2] relaxes an analogous construction using vertex twins for the Laplacian matrix to a set of cousins. We will explain this construction more and generalize it in the following section.…”