The k -maximal hypergraph of commutative rings
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The k-maximal hypergraph of commutative rings K. Selvakumar1 · V. C. Amritha1 Received: 22 December 2019 / Accepted: 22 May 2020 © The Managing Editors 2020
Abstract Let R be a commutative ring with identity, k ≥ 2 a fixed integer and I(R, k) be the set of all k-maximal elements in R. The k-maximal hypergraph associated with R, denoted by Hk (R), is a hypergraph with the vertex set I(R, k) and for distinct k elementsa1 , a2 , . . . , ak in I(R, kk) the set {a1 , a2 , . . . , ak } is an edge of H (R) if and k only if i=1 Rai = R and i=1 Rai = R for all 1 ≤ j ≤ k. In this paper, the i= j
connectedness, diameter and girth of Hk (R) are studied. Moreover, the regularity and coloring of Hk (R) are investigated. Among other things, we characterize all finite commutative rings R for which the k-maximal hypergraph Hk (R) is outerplanar and planar. Keywords Co-maximal graph · k-maximal hypergraph · Connectedness · Girth · Planar graph Mathematics Subject Classification 05C12 · 05C25 · 05C69 · 13A15
1 Glimpse In 1995, Sharma and Bhatwadekar defined a graph on R, with vertices as elements of R, where two distinct vertices x and y are adjacent if and only if Rx + Ry = R. Later, Maimani et al. (2008) studied the graph structure defined by Sharma and Bhatwadekar and named such graph structure co-maximal graph denoted by 2 (R). Eslahchi and Rahimi (2007) have introduced and investigated a graph called the k-zero-divisor hypergraph of a commutative ring. The k-zero-divisor hypergraph of commutative ring has been studied extensively by several authors, see Haouaoui and Benhissi (2012) and Tamizh Chelvam et al. (2015). This motivates us to extend the concept of co-
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V. C. Amritha [email protected] K. Selvakumar [email protected]
1
Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, India
123
Beitr Algebra Geom
maximal graph to k-maximal hypergraph and investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of its associated kmaximal hypergraph. In this paper, we present an algebraic structure called k-maximal hypergraph of a commutative ring R and characterize its related hypergraph called kmaximal hypergraph indicated by Hk (R). Hypergraph theory is a dynamic range in combinatorics, both in direct research, and in application to other zones of science. A hypergraph H is a pair (V (H), E(H)) of disjoint sets, where V (H) is a non empty finite set whose elements are called vertices and the elements of E(H) are non empty subsets of V (H) called edges. The hypergraph H is called k-uniform if every edge e of H is of size k. A path in a hypergraph H is an alternating sequence of distinct vertices and edges of the form v1 , e1 , v2 , e2 , . . . , vk such that vi , vi+1 is in ei for all 1 ≤ i ≤ k − 1. The number of edges of a path is its length. The distance between two vertices x and y of H, denoted by dH (x, y), is the length of the shortest path from x to y. If no such path between x and y exists, we set dH (x, y) = ∞. The greates
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