Overdegrees

Overdegrees refers to a concept in graph theory concerning the maximum degree of a graph. In a simple graph, where no loops or multiple edges exist between the same pair of vertices, the degree of a vertex is the number of edges incident to it. The maximum degree of a graph, denoted as Δ(G), is the highest degree among all vertices in the graph. The term "overdegrees" might colloquially refer to situations where the degree of a vertex exceeds a certain expected or typical value within a specific context, or perhaps when a graph's maximum degree is unusually high compared to its number of vertices or edges.

For instance, in a complete graph $K_n$, every vertex has degree $n-1$, making the maximum degree $n-1$. In contrast, a path graph or a cycle graph has a maximum degree of 2 (for graphs with more than two vertices). The presence of vertices with degrees significantly larger than the average degree can indicate specific structural properties or roles within a network. Such high-degree vertices are often called "hubs" in network analysis. The study of overdegrees, or high-degree vertices, is relevant in understanding the robustness, connectivity, and information flow within complex networks.