On the Semigroup Whose Elements Are Subgraphs of a Complete Graph

Abstract

Let K n be a complete graph on n vertices. Denote by SK n the set of all subgraphs of K n. For each G, H ∈ SK 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 SK n is a semigroup under the ring sum. In this paper, we study Green's relations on SK n and characterize ideals, minimal ideals, maximal ideals, and principal ideals of SK n. Moreover, maximal subsemigroups and a class of maximal congruences are investigated. Furthermore, we prescribe the natural order on SK n and consider minimal elements, maximal elements and covering elements of SK n under this order.