On the Laplacian Coefficients of Bicyclic Graphs

Xinying Pai

doi:10.13189/ujam.2015.030502

Abstract

In this paper, we investigate how the Laplacian coefficients changed after some graph transformations. So, I express some results about Laplacian coefficients ordering of graphs, focusing our attention to the bicyclic graphs. Finally, as an application of these results, we discuss the ordering of graphs based on their Laplacian like energy.

Highlights

The study on the Laplacian coefficient attracts much attention
Some works on Laplacian coefficients can be found in [2,3,4,5,6,7,8]
The Laplacian coefficients ck (G) of a i=1 graph G can be expressed in terms of subtree structures of G by the following result of Kelmans and Chelnokov [6]

Summary

Introduction

The study on the Laplacian coefficient attracts much attention. Some works on Laplacian coefficients can be found in [2,3,4,5,6,7,8]. We determine the largest coefficient among all the bicyclic graphs of order n. Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. Let Bn be the set of all connected bicyclic graphs of order n. Vertices, E edges and L= (G) D(G) − A(G) be its Laplacian matrix. The Laplacian polynomial Φ(G, λ) of G is the characteristic polynomial of its Laplacian matrix. It is well-known that c0 (G ) = 1 , c1 (G ) = 2 E |, cn (G ) = 0 , and cn−1(G) = nτ (G) , where τ (G) is the number of spanning trees of G (see [1])

Transformations and Lemmas

Laplacian Coefficients of Bicyclic Graphs

Main Results

Laplacian-like Energy of Bicyclic Graphs