Degrees and Connectivities of a Graph and Its $δ$-Complement

Abstract

The $δ$-complement $G_δ$ of a graph $G$, introduced in 2022 by Pai et al., is a variant of the graph complement, where two vertices are adjacent in $G_δ$ if and only if they are of the same degree but not adjacent in $G$ or they are of different degrees but adjacent in $G$. In this paper, we provide the Nordhaus-Gaddum-type bounds, in the spirit of Nordhaus and Gaddum (1956), over the maximum degrees, the minimum degrees, the vertex connectivities, and the edge connectivities of a graph and its $δ$-complement. All bounds are attained except for the upper bounds on the product between the minimum degrees of a graph and its $δ$-complement, the vertex connectivities of a graph and its $δ$-complement, and the edge connectivities of a graph and its $δ$-complement.