Distance (signless) Laplacian spectra and energies of two classes of cyclic polyomino chains

Abstract

Let $D\left(G\right)$ and $Tr\left(G\right)$ be the distance matrix and the diagonal matrix of vertex transmissions of a graph G, respectively. The distance Laplacian matrix and the distance signless Laplacian matrix of G are defined as $D^L\left(G\right)=Tr\left(G\right)-D\left(G\right)$ and $D^Q\left(G\right)=Tr\left(G\right)+D\left(G\right)$, respectively. In this paper, we consider the distance Laplacian spectra and the distance signless Laplacian spectra of the linear cyclic polyomino chain $F_n$ and the Möbius cyclic polyomino chain $M_n$. By utilizing the properties of circulant matrices, we give the characteristic polynomials and the eigenvalues for the distance Laplacian matrices and the distance signless Laplacian matrices of the graphs $F_n$ and $M_n$, respectively. Furthermore, we provide the exactly values of the distance Laplacian energy and the distance signless Laplacian energy of the graph $F_n$, and the upper bounds on the distance Laplacian energy and the distance signless Laplacian energy of the graph $M_n$, respectively.