Intersection graph of submodules of a module

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Akbari, S ; Sharif University of Technology

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Type of Document: Article
DOI: 10.1142/S0219498811005452
Abstract:
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
Keywords:
Chromatic number ; Clique number ; Intersection graph
Source: Journal of Algebra and its Applications ; Volume 11, Issue 1 , 2012 ; 02194988 (ISSN)
URL: http://www.worldscientific.com/doi/abs/10.1142/S0219498811005452