Let R be a ring with unity. The idempotent graph G Id (R) of a ring R is an undirected simple graph whose vertices are the set of all the elements of ring R and two vertices x and y are adjacent if and only if x + y is an idempotent element of R. In this paper, we obtain a necessary and sufficient condition on the ring R such that G Id (R) is planar. We prove that G Id (R) cannot be an outerplanar graph. Moreover, we classify all the finite non-local commutative rings R such that G Id (R) is a cograph, split graph and threshold graph, respectively. We conclude that latter two graph classes of G Id (R) are equivalent if and only2010 Mathematics Subject Classification. 05C25.
The inclusion ideal graph In(S) of a semigroup S is an undirected simple graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if either I ⊂ J or J ⊂ I. The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of In(S). In this paper, we investigate the connectedness of In(S). We show that diameter of In(S) is at most 3 if it is connected. We also obtain a necessary and sufficient condition of S such that the clique number of In(S) is n, where n is the number of minimal left ideals of S. Further, various graph invariants of In(S) viz. perfectness, planarity, girth etc. are discussed. For a completely simple semigroup S, we investigate various properties of In(S) including its independence number and matching number. Finally, we obtain the automorphism group of In(S).
Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by Γ ′ (R), is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if x / ∈ Ry and y / ∈ Rx. In this article, we extend some of the results of [24] to an arbitrary ring. In this connection, we derive a closed-form formula of the Wiener index of the cozero-divisor graph of a finite commutative ring R. As applications, we compute the Wiener index of Γ ′ (R), when either R is the product of ring of integers modulo n or a reduced ring. At the final part of this paper, we provide a SageMath code to compute the Wiener index of the cozero-divisor graph of these class of rings including the ring Zn of integers modulo n. et al. [5] studied the cozero-divisor graph associated to the polynomial ring and the ring of power series. Some of the work associated with the cozero-divisor graphs of rings can be found in [3,4,7,11,23,24,25].Over the recent years, the Wiener index of certain graphs associated with rings have been studied by various authors. The Wiener index of the zero divisor graph of the ring Z n of integers modulo n has been studied in [10]. Recently, Selvakumar et al. [28] calculated the Wiener index of the zero divisor graph for a finite commutative ring with unity. The Wiener index of the cozero-divisor graph of the ring Z n has been obtained in [24]. In order to extend the results of [24] to an arbitrary ring, we study the Wiener index of the cozero-divisor graph of a finite commutative 2020 Mathematics Subject Classification. 05C25, 05C50.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.