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
The Journal of Analysis – Springer Journals
Published: Jun 4, 2018
It’s your single place to instantly
discover and read the research
that matters to you.
Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.
All for just $49/month
Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly
Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.
Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.
Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.
All the latest content is available, no embargo periods.
“Hi guys, I cannot tell you how much I love this resource. Incredible. I really believe you've hit the nail on the head with this site in regards to solving the research-purchase issue.”Daniel C.
“Whoa! It’s like Spotify but for academic articles.”@Phil_Robichaud
“I must say, @deepdyve is a fabulous solution to the independent researcher's problem of #access to #information.”@deepthiw
“My last article couldn't be possible without the platform @deepdyve that makes journal papers cheaper.”@JoseServera