Intersection graphs of ideals of rings |
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Authors: | Ivy Chakrabarty Shamik Ghosh TK Mukherjee MK Sen |
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Institution: | aDepartment of Mathematics, Jadavpur University, Kolkata-700 032, India;bDepartment of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata-700 019, India |
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Abstract: | In this paper, we consider the intersection graph G(R) of nontrivial left ideals of a ring R. We characterize the rings R for which the graph G(R) is connected and obtain several necessary and sufficient conditions on a ring R such that G(R) is complete. For a commutative ring R with identity, we show that G(R) is complete if and only if G(Rx]) is also so. In particular, we determine the values of n for which is connected, complete, bipartite, planar or has a cycle. Next, we characterize finite graphs which arise as the intersection graphs of and determine the set of all non-isomorphic graphs of for a given number of vertices. We also determine the values of n for which the graph of is Eulerian and Hamiltonian. |
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Keywords: | Ring Artinian ring Ideal of a ring Intersection graph Connected graph Complete graph Bipartite graph Planar graph Cycle Eulerian graph Hamiltonian graph Unordered factorization |
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