Laplacian spectral characterization of dumbbell graphs and theta graphs

TL;DR

This paper proves that dumbbell graphs and theta graphs are uniquely identified by their Laplacian spectra, contributing to spectral graph theory by characterizing these specific graph classes.

Contribution

It establishes that all dumbbell and theta graphs are determined by their Laplacian spectra, a novel spectral characterization result.

Findings

Dumbbell graphs are uniquely determined by their Laplacian spectra.

Theta graphs are uniquely determined by their Laplacian spectra.

Spectral characterization aids in graph identification and classification.

Abstract

Let $P_n$ and $C_n$ denote the path and cycle on $n$ vertices respectively. The dumbbell graph, denoted by $D_{p,k,q}$, is the graph obtained from two cycles $C_p$, $C_q$ and a path $P_{k+2}$ by identifying each pendant vertex of $P_{k+2}$ with a vertex of a cycle respectively. The theta graph, denoted by $\Theta_{r,s,t}$, is the graph formed by joining two given vertices via three disjoint paths $P_{r}$, $P_{s}$ and $P_{t}$ respectively. In this paper, we prove that all dumbbell graphs as well as theta graphs are determined by their Laplacian spectra.