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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

K. Arathi Bhat1 

G. Sudhakara2 

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

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

Abstract

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,...,Vk} be a partition of V of order k ≥ 1. The k-complement GP k of G with respect to P is defined as follows: For all Vi and Vj in P, i ≠ j, remove the edges between Vi and Vj , 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 GP k ≅ G and is k-co-self complementary (k-co.s.c.) if GP k ≅ Ġ with respect to a partition P of V (G). The mth power of an undirected graph G, denoted by Gm 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 C2 n and the respective partition P of V (C2 n). Finally, we prove that none of the C2 n is k-co-self complementary for any partition P of V (C2 n).

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

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References

[1] K. Arathi Bhat and G. Sudhakara, “Commuting Decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GP 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

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