# Note on generalized Cayley graph of finite rings and its complement

Note on generalized Cayley graph of finite rings and its complement Let R be a finite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph $$\Gamma =\Gamma (R,U(R),U(R))$$ Γ = Γ ( R , U ( R ) , U ( R ) ) is the simple undirected graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u in U(R) such that $$x+uy$$ x + u y is a unit in R. In this paper, we give a necessary and sufficient condition for $$\Gamma$$ Γ to be unicyclic or split or claw-free. Also, we give a necessary and sufficient condition for $$\overline{\Gamma }$$ Γ ¯ to be claw-free or unicyclic or pancyclic. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

# Note on generalized Cayley graph of finite rings and its complement

The Journal of Analysis, Volume 27 (2) – Jun 4, 2018
12 pages

/lp/springer_journal/note-on-generalized-cayley-graph-of-finite-rings-and-its-complement-M9pds0HKK8
Publisher
Springer Journals
Subject
Mathematics; Analysis; Functional Analysis; Abstract Harmonic Analysis; Special Functions; Fourier Analysis; Measure and Integration
ISSN
0971-3611
eISSN
2367-2501
D.O.I.
10.1007/s41478-018-0094-5
Publisher site
See Article on Publisher Site

### Abstract

Let R be a finite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph $$\Gamma =\Gamma (R,U(R),U(R))$$ Γ = Γ ( R , U ( R ) , U ( R ) ) is the simple undirected graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u in U(R) such that $$x+uy$$ x + u y is a unit in R. In this paper, we give a necessary and sufficient condition for $$\Gamma$$ Γ to be unicyclic or split or claw-free. Also, we give a necessary and sufficient condition for $$\overline{\Gamma }$$ Γ ¯ to be claw-free or unicyclic or pancyclic.

### Journal

The Journal of AnalysisSpringer Journals

Published: Jun 4, 2018

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