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

Copyright q 2012 X. Pai and S. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let Φ G, λ det λIn − L G ∑n k 0 −1 ck G λn−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 ck G and investigate a conjecture A. Ilić andM. 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.