On Laplacian and Distance Laplacian Spectra of Generalized Fan Graph & a New Graph Class

TL;DR

This paper analyzes the Laplacian and distance Laplacian spectra of generalized fan graphs and introduces a new graph class, providing spectral characterizations for these structures.

Contribution

It computes the spectra of generalized fan graphs and introduces a new graph class with spectral properties, expanding understanding of graph spectra.

Findings

Spectra of generalized fan graphs are determined.

A new graph class (F_{m,n}) is introduced.

Spectral properties of the new class are characterized.

Abstract

Given a graph $G$, the Laplacian matrix of $G$, $L(G)$ is the difference of the adjacency matrix $A(G)$ and $\text{Deg}(G)$, where $\text{Deg}(G)$ is the diagonal matrix of vertex degrees. The distance Laplacian matrix $D^L({G})$ is the difference of the transmission matrix of $G$ and the distance matrix of $G$. In the given paper, we first obtain the Laplacian and distance Laplacian spectrum of generalized fan graphs. We then introduce a new graph class which is denoted by $\mathcal{NC}(F_{m,n})$. Finally, we determine the Laplacian spectrum and the distance Laplacian spectrum of $\mathcal{NC}(F_{m,n})$.