Intersection graphs of ideals of rings. (English) Zbl 1193.05086

Chakrabarty, Ivy; Ghosh, Shamik; Mukherjee, T. K.; Sen, M. K.

Kaschek, Roland (ed.) et al., Workshop on graph asymmetries, Massey University, Palmerston North, New Zealand, February 23–25, 2005. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 23, 23-32 (2005).

Summary: 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 disconnected 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 \(G(R[x])\) is also so. In particular, we determine the values of \(n\) for which \(G(\mathbb Z_n)\) is connected, complete, bipartite, planar or has a cycle. Next we characterize finite graphs which are the intersection graphs of \(\mathbb Z_n\) and determine the set of all non-isomorphic graphs of \(\mathbb Z_n\) for a given number of vertices. We also determine the values of \(n\) for which the graph of \(\mathbb Z_n\) is Eulerian and Hamiltonian.
[Remark: This paper has later been published in the same form in Discrete Math. 309, No.17, 5381–5392 (2009; Zbl 1193.05087).]
For the entire collection see [Zbl 1109.05008].

MSC:

05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
13M05	Structure of finite commutative rings
13A15	Ideals and multiplicative ideal theory in commutative rings
05C75	Structural characterization of families of graphs
16D25	Ideals in associative algebras

Keywords:

finite rings; Artinian rings; ideals of rings; intersection graph; connected graph; complete graph; bipartite graph; planar graph; cycles; Eulerian graph; Hamiltonian graph; unordered factorizations

Citations:

Zbl 1193.05087

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