On the structure of compact graphs
A simple graph \(G\) is called a compact graph if \(G\) contains no isolated vertices and for each pair \(x\), \(y\) of non-adjacent vertices of \(G\), there is a vertex \(z\) with \(N(x)\cup N(y)\subseteq N(z)\), where \(N(v)\) is the neighborhood of \(v\), for every vertex \(v\) of \(G\). In this...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AGH Univeristy of Science and Technology Press
2017-01-01
|
| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | http://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3747.pdf |
| _version_ | 1818948838916882432 |
|---|---|
| author | Reza Nikandish Farzad Shaveisi |
| author_facet | Reza Nikandish Farzad Shaveisi |
| author_sort | Reza Nikandish |
| collection | DOAJ |
| description | A simple graph \(G\) is called a compact graph if \(G\) contains no isolated vertices and for each pair \(x\), \(y\) of non-adjacent vertices of \(G\), there is a vertex \(z\) with \(N(x)\cup N(y)\subseteq N(z)\), where \(N(v)\) is the neighborhood of \(v\), for every vertex \(v\) of \(G\). In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph \(G\), then the descending chain condition holds for the set of neighbors of \(G\). |
| first_indexed | 2024-12-20T08:53:10Z |
| format | Article |
| id | doaj.art-6495c7d35f994eb1a850086b257b5eef |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-20T08:53:10Z |
| publishDate | 2017-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-6495c7d35f994eb1a850086b257b5eef2022-12-21T19:46:05ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742017-01-01376875886http://dx.doi.org/10.7494/OpMath.2017.37.6.8753747On the structure of compact graphsReza Nikandish0Farzad Shaveisi1Department of Basic Sciences, Jundi-Shapur University of Technology, P.O. Box 64615-334, Dezful, IranDepartment of Mathematics, Faculty of Sciences, Razi University, P.O. Box 67149-67346, Kermanshah, IranA simple graph \(G\) is called a compact graph if \(G\) contains no isolated vertices and for each pair \(x\), \(y\) of non-adjacent vertices of \(G\), there is a vertex \(z\) with \(N(x)\cup N(y)\subseteq N(z)\), where \(N(v)\) is the neighborhood of \(v\), for every vertex \(v\) of \(G\). In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph \(G\), then the descending chain condition holds for the set of neighbors of \(G\).http://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3747.pdfcompact graphvertex degreecycleneighborhood |
| spellingShingle | Reza Nikandish Farzad Shaveisi On the structure of compact graphs Opuscula Mathematica compact graph vertex degree cycle neighborhood |
| title | On the structure of compact graphs |
| title_full | On the structure of compact graphs |
| title_fullStr | On the structure of compact graphs |
| title_full_unstemmed | On the structure of compact graphs |
| title_short | On the structure of compact graphs |
| title_sort | on the structure of compact graphs |
| topic | compact graph vertex degree cycle neighborhood |
| url | http://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3747.pdf |
| work_keys_str_mv | AT rezanikandish onthestructureofcompactgraphs AT farzadshaveisi onthestructureofcompactgraphs |