Evaluating a permutation of the power and complement operations of a graph with respect to its independence numbers

<jats:p> Let <jats:inline-formula> <jats:tex-math>G</jats:tex-math> </jats:inline-formula> be a simple graph and <jats:inline-formula> <jats:tex-math>k\in\mathbb{N}</jats:tex-math> </jats:inline-formula> . Two important graph operations are: the <jats:inline-formula> <jats:tex-math>k</jats:tex-math> </jats:inline-formula> -th graph power of <jats:inline-formula> <jats:tex-math>G</jats:tex-math> </jats:inline-formula> , denoted by <jats:inline-formula> <jats:tex-math>G^{k}</jats:tex-math> </jats:inline-formula> , where <jats:inline-formula> <jats:tex-math>G^{k}</jats:tex-math> </jats:inline-formula> is the graph obtained from <jats:inline-formula> <jats:tex-math>G</jats:tex-math> </jats:inline-formula> by adding an edge between every pair of vertices that have a distance at most <jats:inline-formula> <jats:tex-math>k</jats:tex-math> </jats:inline-formula> , and the complement graph of <jats:inline-formula> <jats:tex-math>G</jats:tex-math> </jats:inline-formula> , denoted by <jats:inline-formula> <jats:tex-math>\overline{G}</jats:tex-math> </jats:inline-formula> . In this paper, we studied the relation between the independence numbers of <jats:inline-formula> <jats:tex-math>\overline{G}^{k}</jats:tex-math> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math>\overline{G^k}</jats:tex-math> </jats:inline-formula> . </jats:p>