Degrees and Connectivities of a Graph and Its $\delta$Complement
Abstract
The $\delta$complement $G_\delta$ of a graph $G$, introduced in 2022 by Pai et al., is a variant of the graph complement, where two vertices are adjacent in $G_\delta$ if and only if they are of the same degree but not adjacent in $G$ or they are of different degrees but adjacent in $G$. In this paper, we provide the NordhausGaddumtype 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 $\delta$complement. All bounds are attained except for the upper bounds on the product between the minimum degrees of a graph and its $\delta$complement, the vertex connectivities of a graph and its $\delta$complement, and the edge connectivities of a graph and its $\delta$complement.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.02507
 arXiv:
 arXiv:2402.02507
 Bibcode:
 2024arXiv240202507S
 Keywords:

 Mathematics  Combinatorics