### Abstract

In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and G^{P}
_{k} , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(G^{P}
_{k} ) is realizable as a graph if and only if P satis-es perfect matching property. For A(G)A(G^{P}
_{k} ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which G^{P}
_{k} and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which G^{P}
_{k} is a graph of rank r and A(G)A(G^{P}
_{k} ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph K_{n1,n2,...,nk} which has a commuting decomposition into a perfect matching and its k-complement.

Original language | English |
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Pages (from-to) | 343-356 |

Number of pages | 14 |

Journal | Special Matrices |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - 01-01-2018 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Geometry and Topology

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## Cite this

_{n1,n2,...,nk}through realization of the product A(G)A(G

^{P}

_{k}).

*Special Matrices*,

*6*(1), 343-356. https://doi.org/10.1515/spma-2018-0028