On the Laplacian coefficients and Laplacian-like energy of unicyclic graphs with \(n\) vertices and \(m\) pendant vertices (Q1952839)

Summary: Let \(\Phi(G, \lambda) = \det(\lambda I_n - L(G)) = \sum^n_{k=0}(-1)^k c_k(G)\lambda^{n-k}\) 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 \textit{A. Ilić} and \textit{M. Ilić} [Linear Algebra Appl. 431, No. 11, 2195--2202 (2009; Zbl 1194.05089)] about the Laplacian coefficients of unicyclic graphs with \(n\) vertices and \(m\) pendant vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with \(n\) vertices and \(m\) pendant vertices.