On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices

Abstract

Let $\mathrm{\Phi }(G,\lambda )=\text{d}\text{e}\text{t}(\lambda I_n-L(G))={\sum }_{k=0}^n(-1{)}^k{c}_k(G){\lambda }^{n-k}$ be the characteristic polynomial of the Laplacian matrix of a graph $G$ of order $n$. In this paper, we give four transforms on graphs that decrease all Laplacian coefficients $c_k(G)$ and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs with $n$ vertices and $m$ pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with $n$ vertices and $m$ pendent vertices.