Some Posets of Unicyclic Graphs Based on Signless Laplacian Coefficients

Abstract

Let $G$ be a graph of order $n$ and let $Q(G,x)=det(xI–Q(G))=\sum _{i=0}^n(-1{)}^i{\zeta }_i(G)x^{n-i}$ be the characteristic polynomial of the signless Laplacian matrix of $G$. We show that the Lollipop graph, $L_{n,3}$, has the maximal $Q$-coefficients, among all unicyclic graphs of order $n$ except $C_n$. Moreover, we determine graphs with minimal $Q$-coefficients, among all unicyclic graphs of order $n$.