On signless Laplacian coefficients of bicyclic graphs

Abstract

Let $G$ be a graph of order $n$ and $Q_G(x)= det(xI-Q(G))= \sum_{i=1}^n (-1)^i \varphi_i x^{n-i}$ 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 $\mathcal{B}(n)$ of all $n$-vertex bicyclic graphs. $\mathcal{B}^1(n)$ denotes all n-vertex bicyclic graphs with at least one odd cycle. We show that $B_n^1$ (obtained from $C_4$ by adding one edge between two non-adjacent vertices and adding $n-4$ pendent vertices at the vertex of degree 3) minimizes all the signless Laplacian coefficients in the set $\mathcal{B}^1(n)$. Moreover, we prove that $B_n^2$ (obtained from $K_{2,3}$ by adding $n-5$ pendent vertices at one vertex of degree 3) has minimum signless Laplacian coefficients in the set $\mathcal{B}^2(n)$ of all $n$-vertex bicyclic graphs with two even cycles.