On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices - Xinying Pai; Sanyang Liu (Journal of Applied Mathematics)
Let Φ(G,λ)=det(λIn-L(G))=∑k=0n(-1)kck(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 and M. 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.