The spectra and the signless Laplacian spectra of graphs with pockets

TL;DR

This paper derives formulas for the signless Laplacian and adjacency spectra of complex graph constructions with pockets, enabling the creation of many cospectral graph pairs.

Contribution

It provides new spectral formulas for graphs with pockets, linking their spectra to those of component graphs, and constructs infinite cospectral graph pairs.

Findings

Signless Laplacian spectra are expressed in terms of component spectra.

Adjacency spectra are derived for graphs with pockets.

Infinite pairs of cospectral graphs are constructed.

Abstract

Let $G[F,V_k,H_v]$ be the graph with $k$ pockets, where $F$ is a simple graph of order $n\geq1$, $V_k=\{v_1,\ldots,v_k\}$ is a subset of the vertex set of $F$ and $H_v$ is a simple graph of order $m\geq2$, $v$ is a specified vertex of $H_v$. Also let $G[F,E_k,H_{uv}]$ be the graph with $k$ edge-pockets, where $F$ is a simple graph of order $n\geq2$, $E_k=\{e_1,\ldots,e_k\}$ is a subset of the edge set of $F$ and $H_{uv}$ is a simple graph of order $m\geq3$, $uv$ is a specified edge of $H_{uv}$ such that $H_{uv}-u$ is isomorphic to $H_{uv}-v$. In this paper, we obtain some results describing the signless Laplacian spectra of $G[F,V_k,H_v]$ and $G[F,E_k,H_{uv}]$ in terms of the signless Laplacian spectra of $F,H_v$ and $F,H_{uv}$, respectively. In addition, we also give some results describing the adjacency spectrum of $G[F,V_k,H_v]$ in terms of the adjacency spectra of $F,H_v$. Finally, as an application of these results, we construct infinitely many pairs of signless Laplacian (resp. adjacency) cospectral graphs.