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 \(n\).