Article in volume
Authors:
Title:
On the non-inverse graph of a group
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 42(2) (2022) 315-325
Received: 2021-04-11 , Revised: 2021-09-27 , Accepted: 2022-04-22 , Available online: 2022-10-05 , https://doi.org/10.7151/dmgaa.1392
Abstract:
Let $(G,\ast)$ be a finite group and $S=\{u \in G | u \neq u^{-1} \}$, then the
inverse graph is defined as a graph whose vertices coincide with $G$ such that
two distinct vertices $u$ and $v$ are adjacent if and only if either
$u \ast v \in S$ or $v \ast u \in S$. In this paper, we introduce a modified
version of the inverse graph, called $i^\ast$-graph associated with a group $G$.
The $i^\ast$-graph is a simple graph with vertex set consisting of elements of $G$
and two vertices $x, y \in Γ$ are adjacent if $x$ and $y$ are not inverses
of each other. We study certain properties and characteristics of this graph.
Some parameters of the $i^\ast$-graph are also determined.
Keywords:
inverse graph, non-inverse graph
References:
- M.R. Alfuraidan and Y.F. Zakariya, Inverse graphs associated with finite groups, Electron. J. Graph Theory Appl. 5 (1) (2017).
https://doi.org/10.5614/ejgta.2017.5.1.14 - D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (2) (1999) 434–447.
https://doi.org/10.1006/jabr.1998.7840 - P.J. Cameron and S. Ghosh, The power graph of a finite group, Discrete Math. 311 (13) (2011) 1220–1222.
https://doi.org/10.1016/j.disc.2010.02.011 - E. Carnia, M. Suyudi, I. Aisah and A.K. Supriatna, A review on eigenvalues of adjacency matrix of graph with cliques, (2017) 040001.
- I. Chakrabarty, S. Ghosh, T.K. Mukherjee and M.K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (17) (2009) 5381–5392.
https://doi.org/10.1016/j.disc.2008.11.034 - P.M. Cohn, Basic Algebra (Springer, New Delhi, 2003).
- F. DeMeyer and L. DeMeyer, Zero divisor graphs of semigroups, J. Algebra 283 (1) (2005) 190–198.
https://doi.org/https://doi.org/10.1016/j.disc.2008.11.034 - T.W. Hungerford, Algebra (Springer, New York, 2008).
- B.L. Johnston and F. Richman, Numbers and Symmetry: An Introduction to Algebra (CRC Press, New York, 1997).
- J.V. Kureethara, Hamiltonian cycle in complete multipartite graphs, Annal. Pure Appl. Math. 13 (2) (2017) 223–228.
https://doi.org/10.22457/apam.v13n2a8 - P. Nasehpour, A generalization of zero-divisor graphs, 2019. arXiv preprint arXiv:1911.06244
- E.B. Vinberg, A Course in Algebra (Universities Press, New Delhi, 2012).
- E.W. Weisstein, Clique Number, From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/CliqueNumber.html
- D.B. West, Introduction to Graph Theory (Pearson, New Jersey, 2001).
Close