谱半径上某些Nordhaus–Gaddum型结果的推广

IF 0.5 4区数学 Q3 MATHEMATICS Journal of Combinatorial Designs Pub Date : 2023-09-21 DOI:10.1002/jcd.21919

Junying Lu, Lanchao Wang, Yaojun Chen

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For <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n </mrow>\n <annotation> $k\\ge 2$</annotation>\n </semantics></math>, a <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-edge decomposition <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>H</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>,</mo>\n \n <mi>…</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>H</mi>\n \n <mi>k</mi>\n </msub>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n <annotation> $({H}_{1},{\\rm{\\ldots }},{H}_{k})$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math> spanning subgraphs such that their edge sets form a <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-partition of the edge set of <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>. 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引用次数: 0

摘要

设G$G$是一个简单图，λ（G）$\lambda（G）$是谱G$G$的半径。对于k≥2$k\ge 2$，k$k$边分解（h1，…，Hk）$({H}_{1} ，｛\rm｛\ldots｝｝，{H}_{k} ）$是k$k$生成子图，使得它们的边集形成G$G$的边集的k$k$-划分。在本文中，我们得到了λ（H1）+…+λ（H k）$\lambda({H}_{1} ）+\，\cdots\，+\lambda({H}_{k} ）$根据H i的集团数${H}_{i} $和G$G$的大小，并讨论什么是k$k$-边分解（h1，…，Hk）$({H}_{1} ，｛\rm｛\ldots｝｝，{H}_{k} ）$可以最大化λ（H1）+…+λ（H k）$\lambda({H}_{1} ）+\cdots\，+\lambda({H}_{k} ）$，当G$G$是一个完全图时。这些结果推广了Nosal、Hong和Shu以及Nikiforov关于k=2$k=2$的谱半径的一些Nordhaus–Gaddum型结果。

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Generalizations of some Nordhaus–Gaddum-type results on spectral radius

Let $G$ be a simple graph and $λ (G)$ the spectral radius of $G$ . For $k \geq 2$ , a $k$ -edge decomposition $(H_{1}, \dots, H_{k})$ is $k$ spanning subgraphs such that their edge sets form a $k$ -partition of the edge set of $G$ . In this paper, we obtain some sharp lower and upper bounds for $λ (H_{1}) + \dots + λ (H_{k})$ in terms of the clique number of $H_{i}$ and the size of $G$ , and discuss what $k$ -edge decomposition $(H_{1}, \dots, H_{k})$ can maximize $λ (H_{1}) + \dots + λ (H_{k})$ when $G$ is a complete graph. These generalize some Nordhaus–Gaddum-type results on spectral radius for $k = 2$ , due to Nosal, Hong and Shu, and Nikiforov.

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来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.