maxdeg

Maxdeg is a term used in graph theory to denote the maximum degree of a graph. For an undirected simple graph G = (V, E), the degree deg(v) of a vertex v is the number of edges incident to v. The maximum degree Δ(G) is the maximum value of deg(v) over all vertices v in V. This value provides a coarse measure of how connected the most connected vertex is. In multigraphs, loops contribute 2 to the degree, so the same definition applies with deg(v) counting incidences accordingly. In directed graphs, one often distinguishes maximum out-degree Δ+(G) and maximum in-degree Δ−(G).

Examples help illustrate the concept: a path on n vertices has Δ = 2 (vertices in the interior have degree 2, endpoints have degree 1). A cycle Cn also has Δ = 2. A complete graph Kn has Δ = n − 1, since every vertex is connected to all others.

Basic properties and uses: the minimum degree δ(G) is always less than or equal to Δ(G), and the average degree is 2|E|/|V|, which is at most Δ(G). The maximum degree bounds the performance of many algorithms; for example, a simple greedy vertex coloring uses at most Δ(G) + 1 colors, and Brooks’ theorem gives tighter bounds: χ(G) ≤ Δ(G) for connected graphs that are neither complete nor an odd cycle, in which case χ(G) = Δ(G) + 1.

In practice, maxdeg influences computational considerations in adjacency representations and in graph processes such as coloring, matching, and degeneracy analyses. Related concepts include the minimum degree δ, the average degree, and degeneracy, which captures the maximum minimum degree over all subgraphs.