具有完美匹配的连通图的无符号拉普拉奇特征值 2 的倍率

IF 1 3区数学 Q3 MATHEMATICS, APPLIED Discrete Applied Mathematics Pub Date : 2024-11-16 DOI:10.1016/j.dam.2024.11.003

Jinxing Zhao , Xiaoxiang Yu

引用次数: 0

摘要

图 G 的无符号拉普拉斯矩阵定义为 Q(G)=D(G)+A(G)，其中 D(G) 是 G 的阶对角矩阵，A(G) 是 G 的邻接矩阵。Q(G) 的特征值 μ 的多重性用 mQ(G,μ) 表示。设 G=C(T1,...Tg) 是一个完美匹配的单环图，其中 Ti 是连接在循环 Cg 的顶点 vi 上的有根树。本文证明，当且仅当 g+t 能被 4 整除时，Q(G) 的特征值为 2，其中 t 是偶数阶 Ti 的个数。本文的另一个主要结果给出了具有完美匹配的连通图 G 的特征，即 mQ(G,2)=θ(G)+1，其中θ(G)=|E(G)|-|V(G)|+1 是 G 的循环数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching

The signless Laplacian matrix of a graph

G

is defined as

Q (G) = D (G) + A (G)

, where

D (G)

is the degree-diagonal matrix of

G

and

A (G)

is the adjacency matrix of

G

. The multiplicity of an eigenvalue

μ

Q (G)

is denoted by

m_{Q} (G, μ)

. Let

G = C (T_{1}, \dots, T_{g})

be a unicyclic graph with a perfect matching, where

T_{i}

is a rooted tree attached at the vertex

v_{i}

of the cycle

C_{g}

. It is proved that

Q (G)

has 2 as an eigenvalue if and only if

g + t

is divisible by 4, where

t

is the number of

T_{i}

of even orders. Another main result of this article gives a characterization for a connected graph

G

with a perfect matching such that

m_{Q} (G, 2) = θ (G) + 1

, where

θ (G) = | E (G) | - | V (G) | + 1

is the cyclomatic number of

G

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来源期刊

Discrete Applied Mathematics 数学-应用数学

CiteScore

2.30

自引率

9.10%

发文量

422

审稿时长

4.5 months

期刊介绍： The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal. Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.

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