On the Laplacian Coefficients and Laplacian-like Energy of Unicyclic Graphs with Fixed Diameter

Abstract

Let $G$ be a graph of order $n$ and let $\mathrm{\Phi }(G,\lambda )=det(\lambda I_n–L(G))=\sum _{k=0}^n(-1{)}^k{c}_k(G){\lambda }^{n-k}$ be the characteristic polynomial of the Laplacian matrix of a graph $G$. In this paper, we identify the minimal Laplacian coefficients of unicyclic graphs with $n$ vertices and diameter $d$. Finally, we characterize the graphs with the smallest and the second smallest Laplacian-like energy among the unicyclic graphs with $n$ vertices and fixed diameter $d$.