Eulerian graphs and automorphisms of a maximal graph

Atul Gaur () and Arti Sharma ()
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Atul Gaur: University of Delhi
Arti Sharma: University of Delhi

Indian Journal of Pure and Applied Mathematics, 2017, vol. 48, issue 2, 233-244

Abstract: Abstract Let R be a commutative ring with identity. Let Γ(R) denote the maximal graph corresponding to the non-unit elements of R, i.e., Γ(R) is a graph with vertices the non-unit elements of R, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of R containing both. In this paper, we have shown that, for any finite ring R which is not a field, Γ(R) is a Euler graph if and only if R has odd cardinality. Moreover, for any finite ring R ≅ R 1×R 2× · · · ×R n, where the R i is a local ring of cardinality p i αi for all i, and the p i’s are distinct primes, it is shown that Aut(Γ(R)) is isomorphic to a finite direct product of symmetric groups. We have also proved that clique(G(R)’) = χ(G(R)’) for any semi-local ring R, where G(R)’ denote the comaximal graph associated to R.

Keywords: Maximal graphs; comaximal graphs; graph automorphisms (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s13226-017-0224-9

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