Prime ideal sum graph of a commutative ring

Abstract

Let [Formula: see text] be a commutative ring with identity. The prime ideal sum graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are 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 study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. The clique number, the chromatic number and the domination number of the prime ideal sum graph for some classes of rings are studied. It is observed that under which condition [Formula: see text] is complete. Moreover, the diameter and the girth of [Formula: see text] are studied.

On the cozero-divisor graphs associated to rings

“…Various algebraic graphs associated to ring structures have been defined and investigated by the researchers (see, [1,4,5,8,9,15,19,20]). Because of the vital role of ideals in the theory of rings, numerous authors have studied the graphs on rings with respect to their ideals such as inclusion ideal graph [2], intersection graphs of ideals [10], ideal-relation graph [14], prime ideal sum graph [23], reduced cozero-divisor graph [35], co-maximal ideal graph [36] etc.…”

“…The comprehensive study of graphs associated with algebraic structures has been of substantial interest due to their applications and their connections to automata theory (see [10,11,12]). Various graphs associated with the rings have been studied in the literature, viz: cozero-divisor graph [2], zero divisor graphs [3], annihilator graph [5], intersection graph of ideals [9], comaximal graph [14], prime ideal sum graph [19] etc.…”

“…The prime ideal sum graph of a commutative ring was introduced by Saha et al [19]. The prime ideal sum graph PIS(R) of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I, J are adjacent if and only if I + J is a prime ideal of R. Authors of [19] studied some graph-theoretic properties of PIS(R) such as the clique number, the chromatic number, the domination number etc.…”

“…The metric dimension and strong metric dimension of the prime ideal sum graph of various classes of rings have been discussed in [1,16]. Various embeddings of the prime ideal sum graphs on surfaces have been investigated in [15]. Moreover, they have studied the prime ideal sum graphs with certain forbidden induced subgraphs such as split, threshold, chordal and cograph.…”

Computing Topological Descriptors of Prime Ideal Sum Graphs of Commutative Rings

“…Studies on these topics in the literature motivated us to study computing different descriptors with respect to a prime ideal sum graph. The prime ideal sum graph of a commutative ring was defined in [30]. For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R, and two distinct vertices I and J are adjacent if and only if I + J is a prime ideal of R. They studied some connections between the graph-theoretic properties of this graph and some algebraic properties of rings (see [30] for more details).…”