On the Connected Power Graphs of Semigroups of Homogeneous Elements of Graded Rings

Ilić-Georgijević, Emil

doi:10.1007/s00009-022-02041-2

Cited by 3 publications

(1 citation statement)

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“…Literature is abound with numerous remarkable results concerning a number of constructions of graphs from rings, semigroups or groups and their applications, including automata theory, see for instance [1,11,19,28,29,30,31,36,43,45] and references therein. The intersection graph of a semigroup was introduced by Bosák [10] in 1964.…”

Section: Introductionmentioning

confidence: 99%

On the intersection ideal graph of semigroups

Baloda,

Kumar

2023

QRS

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The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected, then the diameter of Γ(S) is at most two. Further, we classify the semigroups S in terms of their ideals such that the diameter of Γ(S) is two. We obtain the domination number, independence number, girth and the strong metric dimension of Γ(S). We have also investigated the completeness, planarity and perfectness of Γ(S). We show that if S is a completely simple semigroup, then Γ(S) is weakly perfect. More over, in this article, we give an upper bound of the chromatic number of Γ(S). Finally, if S is the union of n minimal left ideals, then we obtain the metric dimension and the automorphism group of Γ(S).

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