On signless Laplacian coefficients of unicyclic graphs with given matching number

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.