Spectral radius, odd [1,b]-factor and spanning k-tree of 1-binding graphs

IF 1 3区数学 Q1 MATHEMATICS Linear Algebra and its Applications Pub Date : 2024-10-28 DOI:10.1016/j.laa.2024.10.023

Ao Fan , Ruifang Liu , Guoyan Ao

{"title":"Spectral radius, odd [1,b]-factor and spanning k-tree of 1-binding graphs","authors":"Ao Fan , Ruifang Liu , Guoyan Ao","doi":"10.1016/j.laa.2024.10.023","DOIUrl":null,"url":null,"abstract":"<div><div>The binding number <math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> of a graph G is the minimum value of <math><mo>|</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>|</mo><mo>/</mo><mo>|</mo><mi>X</mi><mo>|</mo></math> taken over all non-empty subsets X of <math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> such that <math><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>≠</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math>. A graph G is called 1-binding if <math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>1</mn></math>. Let b be a positive integer. An odd <math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math>-factor of a graph G is a spanning subgraph F such that for each <math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math>, <math><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math> is odd and <math><mn>1</mn><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mi>b</mi></math>. Motivated by the result of Fan, Lin and Lu (2022) [10] on the existence of an odd <math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math>-factor in connected graphs, we first present a tight sufficient condition in terms of the spectral radius for a connected 1-binding graph to contain an odd <math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math>-factor, which generalizes the result of Fan and Lin (2024) [8] on the existence of a 1-factor in 1-binding graphs.</div><div>A spanning k-tree is a spanning tree with the degree of every vertex at most k, which is considered as a connected <math><mo>[</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>]</mo></math>-factor. Inspired by the result of Fan, Goryainov, Huang and Lin (2022) [9] on the existence of a spanning k-tree in connected graphs, we in this paper provide a tight sufficient condition based on the spectral radius for a connected 1-binding graph to contain a spanning k-tree.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524004087","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

The binding number

b (G)

of a graph G is the minimum value of

| N_{G} (X) | / | X |

taken over all non-empty subsets X of

V (G)

such that

N_{G} (X) \neq V (G)

. A graph G is called 1-binding if

b (G) \geq 1

. Let b be a positive integer. An odd

[1, b]

-factor of a graph G is a spanning subgraph F such that for each

v \in V (G)

d_{F} (v)

is odd and

1 \leq d_{F} (v) \leq b

. Motivated by the result of Fan, Lin and Lu (2022) [10] on the existence of an odd

[1, b]

-factor in connected graphs, we first present a tight sufficient condition in terms of the spectral radius for a connected 1-binding graph to contain an odd

[1, b]

-factor, which generalizes the result of Fan and Lin (2024) [8] on the existence of a 1-factor in 1-binding graphs.

A spanning k-tree is a spanning tree with the degree of every vertex at most k, which is considered as a connected

[1, k]

-factor. Inspired by the result of Fan, Goryainov, Huang and Lin (2022) [9] on the existence of a spanning k-tree in connected graphs, we in this paper provide a tight sufficient condition based on the spectral radius for a connected 1-binding graph to contain a spanning k-tree.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

1结合图的谱半径、奇数[1,b]因子和跨k树

图 G 的绑定数 b(G) 是在 V(G) 的所有非空子集 X 上取的|NG(X)|/|X|的最小值，使得 NG(X)≠V(G) 。如果 b(G)≥1，则称图形 G 为 1 约束。图 G 的奇数 [1,b]- 因子是一个跨子图 F，对于每个 v∈V(G)，dF(v) 都是奇数且 1≤dF(v)≤b 。受 Fan、Lin 和 Lu (2022) [10] 关于连通图中存在奇数 [1,b]- 因子的结果的启发，我们首先从谱半径的角度提出了连通的 1 绑定图包含奇数 [1,b]- 因子的严密充分条件，它概括了 Fan 和 Lin (2024) [8] 关于 1 绑定图中存在 1 因子的结果。生成 k 树是指每个顶点的度最多为 k 的生成树，它被视为连通的 [1,k]- 因子。受 Fan、Goryainov、Huang 和 Lin（2022）[9] 关于连通图中存在生成 k 树的结果的启发，我们在本文中提供了一个基于谱半径的连通 1 约束图包含生成 k 树的严密充分条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.

期刊最新文献

On the Colin de Verdière graph number and penny graphs Spectral radius, odd [1,b]-factor and spanning k-tree of 1-binding graphs Editorial Board Editorial Board Around strongly operator convex functions