A method for constructing graphs with the same resistance spectrum

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2024-10-14 DOI:10.1016/j.disc.2024.114284

Si-Ao Xu, Huan Zhou, Xiang-Feng Pan

{"title":"A method for constructing graphs with the same resistance spectrum","authors":"Si-Ao Xu, Huan Zhou, Xiang-Feng Pan","doi":"10.1016/j.disc.2024.114284","DOIUrl":null,"url":null,"abstract":"<div><div>Let G be a simple graph with vertex set <math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> and edge set <math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math>. The resistance distance <math><msub><mrow><mi>R</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math> between two vertices <math><mi>x</mi><mo>,</mo><mi>y</mi></math> of G, is defined to be the effective resistance between the two vertices in the corresponding electrical network in which each edge of G is replaced by a unit resistor. The resistance spectrum <math><mi>RS</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> of a graph G is the multiset of the resistance distances between all pairs of vertices in the graph. This paper presents a novel method for constructing graphs with the same resistance spectrum. It is obtained that for any positive integer k, there exist at least <math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math> graphs with the same resistance spectrum. Furthermore, it is shown that for <math><mi>n</mi><mo>≥</mo><mn>10</mn></math>, there are at least <math><mn>2</mn><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>10</mn><mo>)</mo><mi>p</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo><mo>+</mo><mi>q</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo><mo>)</mo></math> pairs of graphs of order n with the same resistance spectrum, where <math><mi>p</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo></math> and <math><mi>q</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo></math> are the numbers of partitions of the integer <math><mi>n</mi><mo>−</mo><mn>9</mn></math> and simple graphs of order <math><mi>n</mi><mo>−</mo><mn>9</mn></math>, respectively.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114284"},"PeriodicalIF":0.7000,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004151","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let G be a simple graph with vertex set

V (G)

and edge set

E (G)

. The resistance distance

R_{G} (x, y)

between two vertices

x, y

of G, is defined to be the effective resistance between the two vertices in the corresponding electrical network in which each edge of G is replaced by a unit resistor. The resistance spectrum

RS (G)

of a graph G is the multiset of the resistance distances between all pairs of vertices in the graph. This paper presents a novel method for constructing graphs with the same resistance spectrum. It is obtained that for any positive integer k, there exist at least

2^{k}

graphs with the same resistance spectrum. Furthermore, it is shown that for

n \geq 10

, there are at least

2 ((n - 10) p (n - 9) + q (n - 9))

pairs of graphs of order n with the same resistance spectrum, where

p (n - 9)

and

q (n - 9)

are the numbers of partitions of the integer

n - 9

and simple graphs of order

n - 9

, respectively.

查看原文本刊更多论文

构建具有相同电阻谱的图形的方法

假设 G 是一个简单图，具有顶点集 V(G) 和边集 E(G)。G 的两个顶点 x,y 之间的电阻距离 RG(x,y) 定义为两个顶点之间在相应电网络中的有效电阻，其中 G 的每条边都由一个单位电阻代替。图 G 的电阻谱 RS(G) 是图中所有顶点对之间电阻距离的多集。本文提出了一种构建具有相同电阻谱的图的新方法。结果表明，对于任意正整数 k，至少存在 2k 个具有相同电阻谱的图。此外，对于 n≥10，至少存在 2((n-10)p(n-9)+q(n-9)) 对具有相同阻力谱的 n 阶图形，其中 p(n-9) 和 q(n-9) 分别是整数 n-9 和 n-9 阶简单图形的分区数。

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分钟内获得全文求助全文

来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.