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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 thecharacteristic polynomial of the signless Laplacian matrix of a graph $G$. Wegive some transformations of $G$ which decrease all signless Laplaciancoefficients in the set $\mathcal{G}(n,m)$. $\mathcal{G}(n,m)$ denotes alln-vertex unicyclic graphs with matching number $m$. We characterize the graphswhich minimize all the signless Laplacian coefficients in the set$\mathcal{G}(n,m)$ with odd (resp. even) girth. Moreover, we find the extremalgraphs which have minimal signless Laplacian coefficients in the set$\mathcal{G}(n)$ of all $n$-vertex unicyclic graphs with odd (resp. even)girth.

On signless Laplacian coefficients of unicyclic graphs with given matching number