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

Note on generalized Cayley graph of finite rings and its complement J Anal https://doi.org/10.1007/s41478-018-0094-5 ORI G INAL RESEARCH PAPER Note on generalized Cayley graph of ﬁnite rings and its complement 1 1 Tamizh Chelvam Thirugnanam Anukumar Kathirvel Selvaraj Received: 28 January 2018 / Accepted: 18 May 2018 Forum D’Analystes, Chennai 2018 Abstract Let R be a ﬁnite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph C ¼ CðR; UðRÞ; UðRÞÞ 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 þ uy is a unit in R. In this paper, we give a necessary and sufﬁcient condition for C to be unicyclic or split or claw-free. Also, we give a necessary and sufﬁcient condition for C to be claw-free or unicyclic or pancyclic. Keywords Commutative ring  Cayley graphs  Local rings  Unicyclic  Pancyclic Mathematics Subject Classiﬁcation 05C25  05C38  05C75  13M05  16U60 1 Introduction Throughout this paper R denotes a ﬁnite commutative ring with nonzero identity. The subsets Z(R) and U(R) denote the set of all zero-divisors and the multiplicative group 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

, Volume OnlineFirst – Jun 4, 2018
12 pages

/lp/springer_journal/note-on-generalized-cayley-graph-of-finite-rings-and-its-complement-M9pds0HKK8
Publisher
Springer Singapore
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

J Anal https://doi.org/10.1007/s41478-018-0094-5 ORI G INAL RESEARCH PAPER Note on generalized Cayley graph of ﬁnite rings and its complement 1 1 Tamizh Chelvam Thirugnanam Anukumar Kathirvel Selvaraj Received: 28 January 2018 / Accepted: 18 May 2018 Forum D’Analystes, Chennai 2018 Abstract Let R be a ﬁnite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph C ¼ CðR; UðRÞ; UðRÞÞ 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 þ uy is a unit in R. In this paper, we give a necessary and sufﬁcient condition for C to be unicyclic or split or claw-free. Also, we give a necessary and sufﬁcient condition for C to be claw-free or unicyclic or pancyclic. Keywords Commutative ring  Cayley graphs  Local rings  Unicyclic  Pancyclic Mathematics Subject Classiﬁcation 05C25  05C38  05C75  13M05  16U60 1 Introduction Throughout this paper R denotes a ﬁnite commutative ring with nonzero identity. The subsets Z(R) and U(R) denote the set of all zero-divisors and the multiplicative group

### Journal

The Journal of AnalysisSpringer Journals

Published: Jun 4, 2018

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