The 2-switch-degree of a graph

TL;DR

The paper studies the 2-switch-degree, the number of active 2-switches that can be applied to a graph within its degree-class realization graph. It introduces active and inactive vertices, the active space, and a suite of exact, invariant properties and formulas linking the 2-switch-degree to combinatorial subgraph counts and Zagreb indices. Key contributions include a precise decomposition of deg(G) via induced 4-vertex subgraphs, explicit formulas involving $dpe(s)$, $c_4$, $p_4$, $k_3$, and $k_4$, and structural results for forests, trees, and unicyclic graphs. These results connect graph-transformability under 2-switches to classical invariants and offer practical insights for analyzing graph spaces with fixed degree sequences, especially in split graphs and families containing trees and unicyclic graphs.

Abstract

In this work, we delve into the study of the 2-switch-degree of a graph $G$, which is nothing more than the degree of $G$ as a vertex of the realization graph $\mathcal{G}(s)$ associated with the degree sequence $s$ of $G$. We explore the characteristics of active and inactive vertices, the basic properties of the degree, explicit formulas for its computation, and its behavior in specific families of graphs, such as trees and unicyclic graphs.

Figures (2)

• Figure 1: The 11 unlabeled graphs of order 4.
• Figure 2: The realization graph $\mathcal{G}(2^31^2)$.

Theorems & Definitions (90)

• Theorem 1.1: chartrand2010graphs, page 24
• Theorem 1.2: arikati1999realization, page 218
• proof
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4