Intersection graphs of ideals of rings

Chakrabarty, Ivy; Ghosh, Suparna; Mukherjee, Tilottama; Sen, M. K.

doi:10.1016/j.endm.2005.06.104

Cited by 35 publications

(43 citation statements)

References 5 publications

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“…Inspired by his work, Csákány and Pollák [4] in 1969, studied the graph of subgroups of a finite group. Recently, in 2009, the intersection graph of ideals of a ring, was considered by Chakrabarty, Ghosh, Mukherjee and Sen [3]. More work on the intersection graph of groups and rings can also be found in [1,7].…”

Section: Introductionmentioning

confidence: 98%

Intersection Graph of Submodules of a Module

2012

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Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-module M , denoted by G(M ), is defined to be the undirected simple graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have nonzero intersection. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M ). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M ) are determined. We study the clique number and the chromatic number of G(M ). Among other results, it is shown that if G(M ) is a bipartite graph, then G(M ) is a star graph.

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Section: Introductionmentioning

confidence: 98%

Intersection Graph of Submodules of a Module

2012

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“…In [10], Chakrabarty et al have considered the intersection graph G(R) of nontrivial left ideals of a ring R, that is, a graph whose vertices are in a one-to-one correspondence with all nontrivial left ideals of R and two distinct vertices are joined by an edge if and only if the corresponding left ideals of R have a nontrivial intersection. They have characterized the rings R for which the graph G(R) is connected and have obtained several necessary and sufficient conditions on the ring R such that G(R) is complete.…”

Section: A Brief Glance At the Graphs Attached To Other Algebraic Strmentioning

confidence: 99%

Graphs Attached to Rings Revisited

et al. 2011

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In this paper, we discuss some recent results on graphs attached to rings. In particular, we deal with comaximal graphs, unit graphs, and total graphs. We then define the notion of cototal graph and, using this graph, we characterize the rings which are additively generated by their zero divisors. Finally, we glance at graphs attached to other algebraic structures. Mathematics Subject Classification (2010)

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“…There has been a couple of papers devoted to study of intersection graph of algebraic structures (see [3,5]). Also, in [2], the intersection graphs of ideals of rings, is studied.…”

Section: Introductionmentioning

confidence: 99%

The intersection graph of ideals of a poset

Afkhami

Khashyarmanesh

2014

Discrete Math. Algorithm. Appl.

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Let (P, ≤) be a partially ordered set (poset, briefly) with a least element 0. The intersection graph of ideal of P , denoted by G(P ), is a graph whose vertices are all non-trivial ideals of P and two distinct vertices I and J are adjacent if and only if I ∩ J = {0}.In this paper, we study some relations between the algebraic properties of posets and graph-theoretic properties of G(P ). We investigate the connectivity, diameter, girth and planarity of the intersection graph. Also, among the other things, we show that if the clique number of G(P ) is finite, then P is finite too.

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Intersection graphs of ideals of rings

Cited by 35 publications

References 5 publications

Intersection Graph of Submodules of a Module

Intersection Graph of Submodules of a Module

Graphs Attached to Rings Revisited

The intersection graph of ideals of a poset

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