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.

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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.