Graphs that are cospectral for the distance Laplacian

Abstract

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L (G)=T(G)-\mathcal{D} (G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. Several general methods are established for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \cdots \geq |\delta^L_{n}|$, where $\delta^L_{k}$ is the coefficient of $x^k$.

Spectra of Variants of Distance Matrices of Graphs and Digraphs: A Survey

“…Throughout the prior discussion, only the distance matrix has been considered. However, the unimodality of the coefficients of the distance Laplacian characteristic polynomial was established recently [11], and next we establish the unimodality of the coefficients of the distance signless Laplacian and normalized distance Laplacian characteristic polynomials. The distance signless Laplacian characteristic polynomial of G is p…”

“…Next we show that unimodality extends to any positive semidefinite matrix, using the method from [11].…”

“…. , |a t | is unimodal [11]. Denote the eigenvalues of M by μ 1 ≤ • • • ≤ μ n and let c denote the multiplicity of eigenvalue zero for M (with c = 0 signifying zero is not an eigenvalue of M).…”

“…Proof The method used in [11] to prove (3) for independent twins can be used to establish the other eigenvector results. Here we show w = [1, −1, 0, .…”

“…The number of edges in a graph is not preserved by cospectrality for D [27], D L [11], or D L [41] and the diameter of a graph has been shown not to be preserved by D-cospectrality [3] and D L -cospectrality [11]. The girth of a graph is the length of the shortest cycle in the graph.…”