在导出的t路径、t=2,3符号图和t距离符号图上。

IF 1.9 Q2 MULTIDISCIPLINARY SCIENCES MethodsX Pub Date : 2025-06-01 Epub Date: 2025-01-14 DOI:10.1016/j.mex.2025.103160

Deepa Sinha, Sachin Somra

{"title":"在导出的t路径、t=2,3符号图和t距离符号图上。","authors":"Deepa Sinha, Sachin Somra","doi":"10.1016/j.mex.2025.103160","DOIUrl":null,"url":null,"abstract":"<div><div>A signed graph <math><mstyle><mi>Σ</mi></mstyle></math> is a pair <math><mrow><mstyle><mi>Σ</mi></mstyle><mo>=</mo><mo>(</mo><mrow><msup><mrow><mstyle><mi>Σ</mi></mstyle></mrow><mi>u</mi></msup><mo>,</mo><mi>σ</mi></mrow><mo>)</mo><mspace></mspace></mrow></math>that consists of a graph <math><mrow><mo>(</mo><mrow><msup><mrow><mstyle><mi>Σ</mi></mstyle></mrow><mi>u</mi></msup><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow></math> and a sign mapping called signature <math><mi>σ</mi></math> from E to the sign group <math><mrow><mo>{</mo><mrow><mo>+</mo><mo>,</mo><mo>−</mo></mrow><mo>}</mo></mrow></math>. In this paper, we discuss the t-path product signed graph <math><mrow><msub><mover><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow><mo>^</mo></mover><mi>t</mi></msub><mspace></mspace></mrow></math>where vertex set of <math><msub><mover><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow><mo>^</mo></mover><mi>t</mi></msub></math> is the same as that of <math><mstyle><mi>Σ</mi></mstyle></math> and two vertices are adjacent if there is a path of length t, between them in the signed graph <math><mstyle><mi>Σ</mi></mstyle></math>. The sign of an edge in the t-path product signed graph is determined by the product of marks of the vertices in the signed graph <math><mstyle><mi>Σ</mi></mstyle></math>, where the mark of a vertex is the product of signs of all edges incident to it. In this paper, we provide a characterization of <math><mstyle><mi>Σ</mi></mstyle></math> which are switching equivalent to t-path product signed graphs <math><msub><mover><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow><mo>^</mo></mover><mi>t</mi></msub></math> for <math><mrow><mi>t</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math> which are switching equivalent to <math><mstyle><mi>Σ</mi></mstyle></math> and also the negation of the signed graph ŋ<math><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow></math> that are switching equivalent to <math><msub><mover><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow><mo>^</mo></mover><mi>t</mi></msub></math> for <math><mrow><mi>t</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math>. We also characterize signed graphs that are switching equivalent to <math><mi>t</mi></math>-distance signed graph <math><msub><mrow><mo>(</mo><mover><mstyle><mi>Σ</mi></mstyle><mo>¯</mo></mover><mo>)</mo></mrow><mi>t</mi></msub></math> for <math><mrow><mi>t</mi><mo>=</mo><mn>2</mn></mrow></math> where 2-distance signed graph <math><mrow><msub><mrow><mo>(</mo><mover><mstyle><mi>Σ</mi></mstyle><mo>¯</mo></mover><mo>)</mo></mrow><mn>2</mn></msub><mo>=</mo><mrow><mo>(</mo><mrow><msup><mi>V</mi><mo>′</mo></msup><mo>,</mo><msup><mi>E</mi><mo>′</mo></msup><mo>,</mo><msup><mi>σ</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></math> defined as follows: the vertex set is same as the original signed graph <math><mstyle><mi>Σ</mi></mstyle></math> and two vertices <math><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mspace></mspace><mo>∈</mo><msub><mrow><mo>(</mo><mover><mstyle><mi>Σ</mi></mstyle><mo>¯</mo></mover><mo>)</mo></mrow><mn>2</mn></msub></mrow></math>, are adjacent if and only if there exists a distance of length two in <math><mstyle><mi>Σ</mi></mstyle></math>. The edge <math><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><msub><mrow><mo>(</mo><mover><mstyle><mi>Σ</mi></mstyle><mo>¯</mo></mover><mo>)</mo></mrow><mn>2</mn></msub></mrow></math> is negative if and only if all the edges, in all the distances of length two in <math><mstyle><mi>Σ</mi></mstyle></math> are negative otherwise the edge is positive. The t-path network along with these characterizations can be used to develop model for the study of various real life problems communication networks.<ul><li>•<div>t-path product signed graph.</div></li><li>•<div>t-distance signed graph.</div></li></ul></div></div>","PeriodicalId":18446,"journal":{"name":"MethodsX","volume":"14 ","pages":"Article 103160"},"PeriodicalIF":1.9000,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11787706/pdf/","citationCount":"0","resultStr":"{\"title\":\"On derived t-path, t=2,3 signed graph and t-distance signed graph\",\"authors\":\"Deepa Sinha, Sachin Somra\",\"doi\":\"10.1016/j.mex.2025.103160\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>A signed graph <math><mstyle><mi>Σ</mi></mstyle></math> is a pair <math><mrow><mstyle><mi>Σ</mi></mstyle><mo>=</mo><mo>(</mo><mrow><msup><mrow><mstyle><mi>Σ</mi></mstyle></mrow><mi>u</mi></msup><mo>,</mo><mi>σ</mi></mrow><mo>)</mo><mspace></mspace></mrow></math>that consists of a graph <math><mrow><mo>(</mo><mrow><msup><mrow><mstyle><mi>Σ</mi></mstyle></mrow><mi>u</mi></msup><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow></math> and a sign mapping called signature <math><mi>σ</mi></math> from E to the sign group <math><mrow><mo>{</mo><mrow><mo>+</mo><mo>,</mo><mo>−</mo></mrow><mo>}</mo></mrow></math>. In this paper, we discuss the t-path product signed graph <math><mrow><msub><mover><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow><mo>^</mo></mover><mi>t</mi></msub><mspace></mspace></mrow></math>where vertex set of <math><msub><mover><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow><mo>^</mo></mover><mi>t</mi></msub></math> is the same as that of <math><mstyle><mi>Σ</mi></mstyle></math> and two vertices are adjacent if there is a path of length t, between them in the signed graph <math><mstyle><mi>Σ</mi></mstyle></math>. The sign of an edge in the t-path product signed graph is determined by the product of marks of the vertices in the signed graph <math><mstyle><mi>Σ</mi></mstyle></math>, where the mark of a vertex is the product of signs of all edges incident to it. In this paper, we provide a characterization of <math><mstyle><mi>Σ</mi></mstyle></math> which are switching equivalent to t-path product signed graphs <math><msub><mover><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow><mo>^</mo></mover><mi>t</mi></msub></math> for <math><mrow><mi>t</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math> which are switching equivalent to <math><mstyle><mi>Σ</mi></mstyle></math> and also the negation of the signed graph ŋ<math><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow></math> that are switching equivalent to <math><msub><mover><mrow><mo>(</mo><mstyle><mi>Σ</mi></mstyle><mo>)</mo></mrow><mo>^</mo></mover><mi>t</mi></msub></math> for <math><mrow><mi>t</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math>. We also characterize signed graphs that are switching equivalent to <math><mi>t</mi></math>-distance signed graph <math><msub><mrow><mo>(</mo><mover><mstyle><mi>Σ</mi></mstyle><mo>¯</mo></mover><mo>)</mo></mrow><mi>t</mi></msub></math> for <math><mrow><mi>t</mi><mo>=</mo><mn>2</mn></mrow></math> where 2-distance signed graph <math><mrow><msub><mrow><mo>(</mo><mover><mstyle><mi>Σ</mi></mstyle><mo>¯</mo></mover><mo>)</mo></mrow><mn>2</mn></msub><mo>=</mo><mrow><mo>(</mo><mrow><msup><mi>V</mi><mo>′</mo></msup><mo>,</mo><msup><mi>E</mi><mo>′</mo></msup><mo>,</mo><msup><mi>σ</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></math> defined as follows: the vertex set is same as the original signed graph <math><mstyle><mi>Σ</mi></mstyle></math> and two vertices <math><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mspace></mspace><mo>∈</mo><msub><mrow><mo>(</mo><mover><mstyle><mi>Σ</mi></mstyle><mo>¯</mo></mover><mo>)</mo></mrow><mn>2</mn></msub></mrow></math>, are adjacent if and only if there exists a distance of length two in <math><mstyle><mi>Σ</mi></mstyle></math>. The edge <math><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><msub><mrow><mo>(</mo><mover><mstyle><mi>Σ</mi></mstyle><mo>¯</mo></mover><mo>)</mo></mrow><mn>2</mn></msub></mrow></math> is negative if and only if all the edges, in all the distances of length two in <math><mstyle><mi>Σ</mi></mstyle></math> are negative otherwise the edge is positive. The t-path network along with these characterizations can be used to develop model for the study of various real life problems communication networks.<ul><li>•<div>t-path product signed graph.</div></li><li>•<div>t-distance signed graph.</div></li></ul></div></div>\",\"PeriodicalId\":18446,\"journal\":{\"name\":\"MethodsX\",\"volume\":\"14 \",\"pages\":\"Article 103160\"},\"PeriodicalIF\":1.9000,\"publicationDate\":\"2025-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11787706/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"MethodsX\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2215016125000081\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2025/1/14 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q2\",\"JCRName\":\"MULTIDISCIPLINARY SCIENCES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"MethodsX","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2215016125000081","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/1/14 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}

引用次数: 0

摘要

签名图Σ是一对Σ = （Σ u， Σ），由一个图（Σ u， E）和一个从E到符号群{+，-}的称为签名Σ的符号映射组成。本文讨论了t路径积有符号图（Σ） ^ t，其中（Σ） ^ t的顶点集与Σ的顶点集相同，并且在有符号图Σ中，如果两个顶点之间存在长度为t的路径，则它们相邻。t路径积带符号图中的边的符号由带符号图Σ中顶点的标记的乘积决定，其中顶点的标记是与它相关的所有边的符号的乘积。在本文中，我们提供了一个表征Σ，它在t = 2,3时转换等价于t路径积签名图（Σ） ^ t，它转换等价于Σ，同时也提供了对t = 2,3时转换等价于（Σ） ^ t的签名图（Σ）的否定。我们还签署了图形特征,是开关相当于t——远程签名图(Σ¯)t t = 2, 2-distance签名图(Σ¯)2 = (V, E,σ)定义如下:顶点集一样的原始签名图Σ和两个顶点u, V∈(Σ¯)2,相邻当且仅当存在一个距离Σ长度两个。边uv∈（Σ¯）2是负的当且仅当Σ中长度2的所有距离中的所有边都是负的否则边是正的。t-路径网络以及这些特征可以用来建立模型来研究各种现实生活中的通信网络问题。•t路积签名图。•t距离带符号图。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

摘要图片

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

On derived t-path, t=2,3 signed graph and t-distance signed graph

A signed graph

Σ

is a pair

Σ = (Σ^{u}, σ)

that consists of a graph

(Σ^{u}, E)

and a sign mapping called signature

σ

from E to the sign group

{+, -}

. In this paper, we discuss the t-path product signed graph

{\hat{(Σ)}}_{t}

where vertex set of

{\hat{(Σ)}}_{t}

is the same as that of

Σ

and two vertices are adjacent if there is a path of length t, between them in the signed graph

Σ

. The sign of an edge in the t-path product signed graph is determined by the product of marks of the vertices in the signed graph

Σ

, where the mark of a vertex is the product of signs of all edges incident to it. In this paper, we provide a characterization of

Σ

which are switching equivalent to t-path product signed graphs

{\hat{(Σ)}}_{t}

for

t = 2, 3

which are switching equivalent to

Σ

and also the negation of the signed graph ŋ

(Σ)

that are switching equivalent to

{\hat{(Σ)}}_{t}

for

t = 2, 3

. We also characterize signed graphs that are switching equivalent to

t

-distance signed graph

{(\bar{Σ})}_{t}

for

t = 2

where 2-distance signed graph

{(\bar{Σ})}_{2} = (V^{'}, E^{'}, σ^{'})

defined as follows: the vertex set is same as the original signed graph

Σ

and two vertices

u, v \in {(\bar{Σ})}_{2}

, are adjacent if and only if there exists a distance of length two in

Σ

. The edge

u v \in {(\bar{Σ})}_{2}

is negative if and only if all the edges, in all the distances of length two in

Σ

are negative otherwise the edge is positive. The t-path network along with these characterizations can be used to develop model for the study of various real life problems communication networks.