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 \(\Phi(G, \lambda) = \det(\lambda I_n – L(G)) = \sum_{k=0}^{n}(-1)^k c_k(G) \lambda^{n-k}\) 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 \(d\).