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.
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.
Let [Formula: see text] be a commutative ring with unity. The prime ideal sum graph of the ring [Formula: see text] is a simple undirected graph whose vertex set is the set of nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we characterize all the finite non-local commutative rings whose prime ideal sum graph is of genus [Formula: see text].
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