The Distance Laplacian and Distance Signless Laplacian Spectra of $\mathcal{C}$-Graphs

Source abstract

Mandal and Mehatari (\emph{Comp.\ Appl.\ Math.}, 2025) introduced the class $\mathcal{C}$ of cographs generated by a finite creation sequence $(α_1,\dots,α_m)$ of natural numbers, and derived the inertia, an extended eigenvalue-free interval, and the exact characteristic polynomial for the \emph{adjacency} matrix of such graphs. In this note we develop the parallel theory for the \emph{distance Laplacian} matrix $D^L(G)$ and \emph{distance signless Laplacian} matrix $D^Q(G)$. It is shown that $(0,α_{\min}) \cup (n-α_{\min}, n)$, $(0,n) \ \cup\ (n,\,n+α_{\min}) \ \cup\ (2n-α_{\min},\,2n)$, $\big(2\Tr_{\min},\ μ_{\max}(D^Q(G))\big)$ are the eigenvalue-free intervals of the Laplacian, distance Laplacian, and distance signless Laplacian matrices respectively for the said class of graphs.