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

Let be the characteristic polynomial of the Laplacian matrix of a graph of order . In this paper, we give four transforms on graphs that decrease all Laplacian coefficients and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs with vertices and pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with vertices and pendent vertices.