The 2-switch-degree of a graph

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.