Bicyclic graphs with maximum sum of the two largest Laplacian eigenvalues

Jianxi Li,Sa Rula,Yirong Zheng,An Chang

doi:10.1186/s13660-016-1235-5

Jianxi Li, Sa Rula + Show 2 more

Open Access

https://doi.org/10.1186/s13660-016-1235-5

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Abstract

Let G be a simple connected graph and $S_{2}(G)$ be the sum of the two largest Laplacian eigenvalues of G. In this paper, we determine the bicyclic graph with maximum $S_{2}(G)$ among all bicyclic graphs of order n, which confirms the conjecture of Guan et al. (J. Inequal. Appl. 2014:242, 2014) for the case of bicyclic graphs.

Highlights

Let G = (V (G), E(G)) be a simple connected graph with vertex set V (G) = {v, v, . . . , vn}.The numbers of its vertices and edges are denoted by n(G) and m(G)
The eigenvalues of L(G) are called the Laplacian eigenvalues of G and are denoted by μ (G) ≥ μ (G) ≥ · · · ≥ μn(G), which are always enumerated in non-increasing order and repeated according to their multiplicity
Let G be the union of some disjoint graphs G, G, . . . , Gr, where Gi (i ∈ {, . . . , r}) is a tree or an unicyclic graph of order ni which is not isomorphic to Gni,ni (Gn,n is the unicyclic graph shown in Figure )

Summary

Introduction

The numbers of its vertices and edges are denoted by n(G) and m(G) (or n and m for short). For a vertex v ∈ V (G), let N(v) be the set of all neighbors of v in G. Denote by (G) (or for short) the maximum degree of G. D(vn)) is the diagonal matrix of vertex degrees of G. We use the notation In for the identity matrix of order n and denote by φ(G, x) = det(xIn – L(G)) the Laplacian characteristic polynomial of G. The eigenvalues of L(G) are called the Laplacian eigenvalues of G and are denoted by μ (G) ≥ μ (G) ≥ · · · ≥ μn(G) (or μ ≥ μ ≥ · · · ≥ μn for short), which are always enumerated in non-increasing order and repeated according to their multiplicity.

Note that each row sum of

Laplacian eigenvalues of

If n and n