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

Abstract

Mandal and Mehatari (\emph{Comp.\ Appl.\ Math.}, 2025) introduced the class $\mathcal{C}$ of cographs generated by a finite creation sequence $({\alpha }_1,\dots ,{\alpha }_m)$ 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 $(0,{\alpha }_{\min })\cup (n-{\alpha }_{\min },n)$, $(0,n)\cup (n,n+{\alpha }_{\min })\cup (2n-{\alpha }_{\min },2n)$, $\big(2\Tr_{\min},\ μ_{\max}(D^Q(G))\big)$ are the eigenvalue-free intervals of the Laplacian, distance Laplacian, and distance signless Laplacian matrices respectively for the said class of graphs.