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.
Highlights
Introduction and PreliminariesOne of several ways to study the algebraic structures in mathematics is to consider the relations between graph theory and semigroup theory known as algebraic graph theory
It is a branch of mathematics concerning the study of graphs in connection to semigroup theory in which algebraic methods are applied to problems about graphs
We present the characterizations of ideals, minimal ideals, maximal ideals, and principal ideals of SKn
Summary
One of several ways to study the algebraic structures in mathematics is to consider the relations between graph theory and semigroup theory known as algebraic graph theory. There are no authors considering the properties of semigroups which are constructed from graphs. We construct a new semigroup from a complete graph and study some valuable properties of such a semigroup. For each a nonempty subset A of Xn , denote by φ A a graph with a vertex set V (φ A ) = A and. In order to consider the rank of SKn , we shall denote by H [e] an induced subgraph of H induced by e where e ∈ E( H ).
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