On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs withnVertices andmPendent Vertices

Xinying Pai,Sanyang Liu

doi:10.1155/2012/404067

Abstract

LetΦ(G,λ)=det(λIn-L(G))=∑k=0n(-1)kck(G)λn-kbe the characteristic polynomial of the Laplacian matrix of a graphGof ordern. In this paper, we give four transforms on graphs that decrease all Laplacian coefficientsck(G)and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs withnvertices andmpendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs withnvertices andmpendent vertices.

Highlights

Let G V, E be a simple undirected graph with n vertices and |E| edges and, let L G D G − A G be its Laplacian matrix
The Laplacian polynomial of G is the characteristic polynomial of its Laplacian matrix
I⊆{1,2,...,n−1},|I| k i∈I Journal of Applied Mathematics is a symmetric polynomial of order n − 1

Summary

Introduction

Let G V, E be a simple undirected graph with n vertices and |E| edges and, let L G D G − A G be its Laplacian matrix. Mohar 2 proved that among all trees of order n, the kth Laplacian coefficients ck G are largest when the tree is a path and are smallest for stars. Stevanovicand Ilic 3 showed that among all connected unicyclic graphs of order n, the kth Laplacian coefficients ck G are largest when the graph is a cycle Cn and smallest when the graph is an Sn with an additional edge between two of its pendent vertices, where Sn is a star of order n. We determine the smallest kth Laplacian coefficients ck G among all unicyclic graphs with n vertices and m pendent vertices. Motivated by the results in 3, 4, 9–12 concerning the minimal Laplacian coefficients and Laplacian-like energy of some graphs and the minimal molecular graph energy of unicyclic graphs with n vertices and m pendent vertices, this paper will characterize the unicyclic graphs with n vertices and m pendent vertices, which minimize Laplacian-like energy

Transformations and Lemmas

Main Results

Laplacian-Like Energy of Unicyclic Graphs with m Pendent Vertices