On signless Laplacian coefficients of unicyclic graphs with given matching number

TL;DR

This paper investigates how to transform unicyclic graphs to minimize their signless Laplacian coefficients, characterizing extremal graphs based on matching number and girth.

Contribution

It introduces transformations that reduce signless Laplacian coefficients and characterizes extremal graphs with minimal coefficients for various girth and matching number conditions.

Findings

Identifies transformations decreasing all signless Laplacian coefficients.

Characterizes graphs minimizing coefficients for fixed matching number and girth.

Finds extremal graphs with minimal coefficients among all unicyclic graphs.

Abstract

Let $G$ be an unicyclic graph of order $n$ and let $Q_G(x)= det(xI-Q(G))={matrix} \sum_{i=1}^n (-1)^i \varphi_i x^{n-i}{matrix}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\mathcal{G}(n,m)$. $\mathcal{G}(n,m)$ denotes all n-vertex unicyclic graphs with matching number $m$. We characterize the graphs which minimize all the signless Laplacian coefficients in the set $\mathcal{G}(n,m)$ with odd (resp. even) girth. Moreover, we find the extremal graphs which have minimal signless Laplacian coefficients in the set $\mathcal{G}(n)$ of all $n$-vertex unicyclic graphs with odd (resp. even) girth.