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

Supakorn Srisawat, Panupong Vichitkunakorn

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.

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

Abstract

The -complement of a graph , introduced in 2022 by Pai et al., is a variant of the graph complement, where two vertices are adjacent in if and only if they are of the same degree but not adjacent in or they are of different degrees but adjacent in . 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 -complement. All bounds are attained except for the upper bounds on the product between the minimum degrees of a graph and its -complement, the vertex connectivities of a graph and its -complement, and the edge connectivities of a graph and its -complement.
Paper Structure (4 sections, 13 theorems, 32 equations)

This paper contains 4 sections, 13 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Nordhaus-Gaddum-Type Relations
  4. Conclusion and Discussion

Key Result

Theorem 1

Let $G$ be a graph of $n$ vertices. Then, and Moreover, the bounds are sharp for all $n$.

Theorems & Definitions (24)

  • Theorem 1: noga
  • Theorem 2: xu2
  • Theorem 3: alavi
  • Theorem 4: alavi
  • Definition 5
  • Theorem 6: panupong
  • Theorem 7
  • proof
  • Theorem 8
  • proof
  • ...and 14 more