Nordhaus-Gaddum-Type Relations Over Invariants of a Graph and Its ?-Complement

Abstract

Let $G$ be a graph. The $\delta$-complement of $G$, denoted by $G_{\delta}$, was defined by Pai et al. (2022) to be a variation of a graph complement where two vertices are adjacent in $G_\delta$ if and only if they are of the same degree but not adjacent in $G$ or they are of different degrees but adjacent in $G$.We provide the sum and the product bounds, in the spirit of Nordhaus and Gaddum (1956) over several invariants of a graph and its $\delta$-complement: minimum degrees, maximum degrees, girths, diameters, radii, domination numbers, vertex connectivities, and edge connectivities.We also use Sagemath (2022) to compute the results on the graphs of order at most 10.??? $G$ ???????? ??????????????? $\delta$ ???????????? $G_{\delta}$ ???????? Pai ?????? (2022) ???????????????????????????????????????????????????? ? ???????????????? $G_{\delta}$ ????????????????????????????????????????????????????????????? $G$ ??????????????????????????????????????????????????? $G$????????????????????????????????????????????????? Nordhaus ??? Gaddum (1956) ???????????????????????????? ? ????????????????????????? $\delta$-complement ???????????????????????? ?????????????? ????????? ??????????? ????? ????????? ????????????????????????? ????????????????????????????????????????????? Sagemath (2022) ?????????????????????????????????????????????? 10 ???