On the existence of semigraphs and complete semigraphs with given parameters

E. Sampathkumar has generalized a graph to a semigraph by allowing an edge to have more than two vertices. Like in the case of graphs, a complete semigraph is a semigraph in which every two vertices are adjacent to each other. In this article, we have generalized a problem noted by Gauss in 1796 abo...

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Autores principales:	Jyoti Shetty, G. Sudhakara, Vinay Madhusudanan
Formato:	article
Lenguaje:	EN
Publicado:	Elsevier 2021
Materias:	Complete semigraph Triangular numbers Vertex degree Engineering (General). Civil engineering (General) TA1-2040
Acceso en línea:	https://doaj.org/article/6539e4e600cc43f7bdc829b890084089
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spelling	oai:doaj.org-article:6539e4e600cc43f7bdc829b8900840892021-11-22T04:21:26ZOn the existence of semigraphs and complete semigraphs with given parameters2090-447910.1016/j.asej.2021.04.002https://doaj.org/article/6539e4e600cc43f7bdc829b8900840892021-12-01T00:00:00Zhttp://www.sciencedirect.com/science/article/pii/S2090447921001611https://doaj.org/toc/2090-4479E. Sampathkumar has generalized a graph to a semigraph by allowing an edge to have more than two vertices. Like in the case of graphs, a complete semigraph is a semigraph in which every two vertices are adjacent to each other. In this article, we have generalized a problem noted by Gauss in 1796 about triangular numbers and shown that it is the deciding factor of when a semigraph is complete.Let P be a set with p elements and {E1,E2,…,Eq} be a collection of subsets of P with ⋃i=1qEi=P. We derive an expression for the maximum value of the difference ∑j=1k\|Eij\|-⋃i=1kEij for 2⩽k⩽q, where every two of the sets in the collection can have at most one element in common. We show that this result helps in answering the question of whether there exists a semigraph on the vertex set P having edges {e1,e2,…,eq}, where the set Ei is the set of vertices on the edge ei,1⩽i⩽q. Combining the above two results, we characterize a complete semigraph.Jyoti ShettyG. SudhakaraVinay MadhusudananElsevierarticleComplete semigraphTriangular numbersVertex degreeEngineering (General). Civil engineering (General)TA1-2040ENAin Shams Engineering Journal, Vol 12, Iss 4, Pp 4119-4124 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Complete semigraph Triangular numbers Vertex degree Engineering (General). Civil engineering (General) TA1-2040
spellingShingle	Complete semigraph Triangular numbers Vertex degree Engineering (General). Civil engineering (General) TA1-2040 Jyoti Shetty G. Sudhakara Vinay Madhusudanan On the existence of semigraphs and complete semigraphs with given parameters
description	E. Sampathkumar has generalized a graph to a semigraph by allowing an edge to have more than two vertices. Like in the case of graphs, a complete semigraph is a semigraph in which every two vertices are adjacent to each other. In this article, we have generalized a problem noted by Gauss in 1796 about triangular numbers and shown that it is the deciding factor of when a semigraph is complete.Let P be a set with p elements and {E1,E2,…,Eq} be a collection of subsets of P with ⋃i=1qEi=P. We derive an expression for the maximum value of the difference ∑j=1k\|Eij\|-⋃i=1kEij for 2⩽k⩽q, where every two of the sets in the collection can have at most one element in common. We show that this result helps in answering the question of whether there exists a semigraph on the vertex set P having edges {e1,e2,…,eq}, where the set Ei is the set of vertices on the edge ei,1⩽i⩽q. Combining the above two results, we characterize a complete semigraph.
format	article
author	Jyoti Shetty G. Sudhakara Vinay Madhusudanan
author_facet	Jyoti Shetty G. Sudhakara Vinay Madhusudanan
author_sort	Jyoti Shetty
title	On the existence of semigraphs and complete semigraphs with given parameters
title_short	On the existence of semigraphs and complete semigraphs with given parameters
title_full	On the existence of semigraphs and complete semigraphs with given parameters
title_fullStr	On the existence of semigraphs and complete semigraphs with given parameters
title_full_unstemmed	On the existence of semigraphs and complete semigraphs with given parameters
title_sort	on the existence of semigraphs and complete semigraphs with given parameters
publisher	Elsevier
publishDate	2021
url	https://doaj.org/article/6539e4e600cc43f7bdc829b890084089
work_keys_str_mv	AT jyotishetty ontheexistenceofsemigraphsandcompletesemigraphswithgivenparameters AT gsudhakara ontheexistenceofsemigraphsandcompletesemigraphswithgivenparameters AT vinaymadhusudanan ontheexistenceofsemigraphsandcompletesemigraphswithgivenparameters
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On the existence of semigraphs and complete semigraphs with given parameters

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