Signless Laplacian spectral analysis of a class of graph joins

TL;DR

This paper proves that certain classes of graph joins are uniquely identified by their signless Laplacian spectra and explicitly computes these spectra, advancing spectral graph theory understanding.

Contribution

It establishes that specific graph join structures are determined by their signless Laplacian spectra and provides explicit spectral calculations for these graphs.

Findings

Graphs of the form $K_1 abla (C_s old qK_2)$ are DQS for sufficiently large vertices.

Graphs combining multiple cycles and $K_2$ components are DQS when sufficiently large.

Explicit signless Laplacian spectra are derived for these graph classes.

Abstract

A graph is said to be determined by its signless Laplacian spectrum (abbreviated as DQS) if no other non-isomorphic graph shares the same signless Laplacian spectrum. In this paper, we establish the following results: (1). Every graph of the form $K_1 \vee (C_s \cup qK_2)$, where $q \ge 0$, $s \ge 3$, and the number of vertices is at least $16$, is DQS; (2). Every graph of the form $K_1 \vee (C_{s_1} \cup C_{s_2} \cup \cdots \cup C_{s_t} \cup qK_2)$, where $t \ge 2$, $q \ge 0$, $s_i \ge 3$, and the number of vertices is at least $52$, is DQS. Here, $K_n$ and $C_n$ denote the complete graph and the cycle of order $n$, respectively, while $\cup$ and $\vee$ represent the disjoint union and the join of graphs. Moreover, the signless Laplacian spectrum of the graphs under consideration is computed explicitly.