## Services on Demand

## Journal

## Article

## Indicators

- Cited by SciELO
- Access statistics

## Related links

- Cited by Google
- Similars in SciELO
- Similars in Google

## Share

## Proyecciones (Antofagasta)

##
*Print version* ISSN 0716-0917

### Proyecciones (Antofagasta) vol.41 no.3 Antofagasta June 2022

#### http://dx.doi.org/10.22199/issn.0717-6279-4638

Artículos

Powers of cycle graph which are *k*-self complementary and *k*-co-self complementary

^{1}Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal Karnataka, 576104 India. e-mail: arathi.bhat@manipal.edu

^{2}Department of Mathematics, Manipal Institute of Technology Manipal Academy of Higher Education, Manipal Karnataka, 576104 India. e-mail: sudhakara.g@manipal.edu

E. Sampath Kumar and L. Pushpalatha [4] introduced a generalized version of complement of a graph with respect to a given partition of its vertex set. Let G = (V,E) be a graph and P = {V₁, V₂,...,V_{k}} be a partition of V of order k ≥ 1. The k-complement G^{P}
_{k} of G with respect to P is defined as follows: For all V_{i} and V_{j} in P, i ≠ j, remove the edges between V_{i} and V_{j} , and add the edges which are not in G. Analogues to self complementary graphs, a graph G is k-self complementary (k-s.c.) if G^{P}
_{k} ≅ G and is k-co-self complementary (k-co.s.c.) if G^{P}
_{k} ≅ Ġ with respect to a partition P of V (G). The m^{th} power of an undirected graph G, denoted by G^{m} is another graph that has the same set of vertices as that of G, but in which two vertices are adjacent when their distance in G is at most m. In this article, we study powers of cycle graphs which are k-self complementary and k-co-self complementary with respect to a partition P of its vertex set and derive some interesting results. Also, we characterize k-self complementary C^{2}
_{n} and the respective partition P of V (C^{2}
_{n}). Finally, we prove that none of the C^{2}
_{n} is k-co-self complementary for any partition P of V (C^{2}
_{n}).

**Keywords and phrases: **k-complement; k(i)-complement; k-self complementary;k-co-self complementary; powers of cycle graph

References

[1] K. Arathi Bhat and G. Sudhakara, “Commuting Decomposition of K_{n1,n2,...,nk} through realization of the product A(G)A(G^{P}
_{k})”, *Special Matrices*, vol. 6, pp. 343-356, 2018.
[ Links ]

[2] K. Arathi Bhat and G. Sudhakara, “Commuting Graphs and their Generalized Complements”, *Malaysian Journal of Mathematical Sciences*, vol. 12, no. 1, pp. 63-84, 2018.
[ Links ]

[3] D. B. West, *Introduction to Graph Theory*. Prentice Hall, 1996.
[ Links ]

[4] E. Sampath Kumar and L. Pushpalatha, “Complement of a graph a generalization”, *Graphs and Combinatorics*, vol. 14, pp. 377-392, 1998.
[ Links ]

[5] E. Sampath Kumar , L. Pushpalatha, Venkatachalam and Pradeep G Bhat, “Generalized complements of a graph”, *Indian Journal of Pure and Applied Mathematics*, vol. 29, no. 6, pp. 625-639, 1998.
[ Links ]

[6] G. Sudhakara, “Wheels, Cages and Cubes,” in Number Theory and Discrete Mathematics, A. K. Agarwal, B. C. Berndt, C. F. Krattenthaler, G. L. Mullen, K. Ramachandra, and M. Waldschmidt, Eds. Basel: Birkhäuser, 2002, pp. 251-259. Doi: 10.1007/978-3-0348-8223-1_25 [ Links ]

Received: December 30, 2020; Accepted: December 30, 2021