The Signless Laplacian Coefficients and the Incidence Energy of Unicyclic Graphs with given Pendent Vertices

Abstract

Let $\mathcal{U}_n^{r}$ be the set of unicyclic graphs with $n$ vertices and $r$ pendent vertices, where $n\geq 5$ and $r\geq 1$. We consider the signless Laplacian coefficients and the incidence energy in $\mathcal{U}_n^{r}$.  To obtain our results,  we introduce two new graph transforms that preserve order, size and the number of the pendent vertices, but decrease the matching number of the subdivision graphs and the signless Laplacian coefficients of the graphs considered. Firstly,  the graph with the minimum signless Laplacian coefficients and the minimum incidence energy is obtained in  $\mathcal{U}_{n,g}^{r}$ with odd $g \geq 3$ and $n\geq g+2$, where $\mathcal{U}_{n,g}^{r}$ is the subset of $\mathcal{U}_n^{r}$ in which every graph has girth $g\geq 3$.  Secondly, in $\mathcal{U}_n^{r}$ with $r\geq 2$ and odd $g \geq 5$, a class of graphs with girth 4 having the minimal signless Laplacian coefficients and the minimal incidence energies are obtained.