Laplacian coefficients of unicyclic graphs with the number of leaves and girth

Jie Zhang,Xiao-Dong Zhang

doi:10.2298/aadm140715008z

Jie Zhang, Xiao-Dong Zhang

Open Access

https://doi.org/10.2298/aadm140715008z

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Abstract

Motivated by Ilic and Ilic?s conjecture [A. Ilic, M. Ilic, Laplacian coefficients of trees with given number of leaves or vertices of degree two, Linear Algebra Appl., 431(2009)2195-2202.], we investigate properties of the minimal elements in the partial set (Ugn,l,?) of the Laplacian coefficients, where Ug n,l denote the set of n-vertex unicyclic graphs with the number of leaves l and girth g. These results are used to disprove their conjecture. Moreover, the graphs with minimum Laplacian-like energy in Ug n,l are also studied.

Highlights

Let G = (V, E) be a simple graph with n = |V | vertices and m edges and L(G) = D(G) − A(G) be its Laplacian matrix, where A(G) and D(G) are its adjacency and degree diagonal matrices, respectively
This paper is organized as follows: in Section 2, we investigate some properties of minimal elements in the poset Ung,l
[5] Let v be a vertex of a nontrivial connected graph G and for nonnegative integers p and q, let G(p, q) denote the graph obtained from G by adding two pendent paths of lengths p and q at v, respectively, p ≥ q ≥ 1

Summary

INTRODUCTION

Laplacian coefficients, Laplacian-like energy, balanced starlike tree. Ilic [4] and Zhang et al [17] investigated ordering trees by the Laplacian coefficients. Stevanovicand Ilic [13] investigated and characterized the minimum and maximum elements in the poset of unicyclic graphs of order n with. Tan [15] proved that the poset of unicyclic graphs of order n and fixed matching number with has only one minimal element. He and Shan [3] studied the properties of the poset of bicyclic graphs of order n.

Denote by

Then their Laplacian characteristic polynomials are

Hence it follows that