On the Semigroup Whose Elements Are Subgraphs of a Complete Graph

Yanisa Chaiya, Chollawat Pookpienlert, Nuttawoot Nupo and Sayan Panma
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Yanisa Chaiya: Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University (Rangsit Campus), Pathum Thani 12121, Thailand
Chollawat Pookpienlert: Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Nuttawoot Nupo: Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Sayan Panma: Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Mathematics, 2018, vol. 6, issue 5, 1-10

Abstract: Let K n be a complete graph on n vertices. Denote by S K n the set of all subgraphs of K n . For each G , H ∈ S K n , the ring sum of G and H is a graph whose vertex set is V ( G ) ∪ V ( H ) and whose edges are that of either G or H , but not of both. Then S K n is a semigroup under the ring sum. In this paper, we study Green’s relations on S K n and characterize ideals, minimal ideals, maximal ideals, and principal ideals of S K n . Moreover, maximal subsemigroups and a class of maximal congruences are investigated. Furthermore, we prescribe the natural order on S K n and consider minimal elements, maximal elements and covering elements of S K n under this order.

Keywords: complete graph; Green’s relations; ideal; natural order; maximal subsemigroup; maximal congruence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
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