Non co-maximal graph of subgroups of an abelian group

Abstract

For an abelian group G of finite order, the non co-maximal graph of subgroups of G, denoted by NC(G), is a graph whose vertices are non-trivial proper subgroups of G and two distinct vertices S and T are adjacent if and only if \(ST \ne G\). In this article, we study the interdisciplinary relation between group theoretic properties and graph theoretic properties of non co-maximal graph. The role of cyclic group is one of the key components in this discussion. We also emphasis on the concept of the maximal subgroups of the groups for depiction of the corresponding graphs. Almost all graph theoretic insights are taken into consideration for the developments of this graph. We investigate completeness, emptiness, connectedness, diameter, girth in the second section of this paper. The clique number, independence number, domination number, vertex chromatic number are found in the third section. Planarity, weakly perfect character are interpreted in the fourth section. In the fifth section, we discuss the concept of traversability of NC(G).