## 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 G^{P}_{k} of G with respect to P is defined as follows: For all V_{i} and V_{j} in P, i 6= 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 (Formula Presented) and is k-co-self complementary (k-co.s.c.) if (Formula Presented) 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 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} ).

Original language | English |
---|---|

Pages (from-to) | 715-732 |

Number of pages | 18 |

Journal | Proyecciones |

Volume | 41 |

Issue number | 3 |

DOIs | |

Publication status | Published - 06-2022 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)