Opuscula Math. 37, no. 6 (2017), 875-886
http://dx.doi.org/10.7494/OpMath.2017.37.6.875

Opuscula Mathematica

# On the structure of compact graphs

Reza Nikandish

Abstract. 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 $$G$$.

Keywords: compact graph, vertex degree, cycle, neighborhood.

Mathematics Subject Classification: 05C07, 05C38, 68R10.

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• Reza Nikandish
• Department of Basic Sciences, Jundi-Shapur University of Technology, P.O. Box 64615-334, Dezful, Iran
• Department of Mathematics, Faculty of Sciences, Razi University, P.O. Box 67149-67346, Kermanshah, Iran
• Communicated by Mirko Horňák.
• Accepted: 2017-02-15.
• Published online: 2017-09-28.