log2diameter

Log2diameter is the base-2 logarithm of the diameter of a graph. For a simple undirected graph G = (V,E), the diameter Diam(G) is the maximum distance between any two vertices, where distance is the length of a shortest path. If G is disconnected, Diam(G) is undefined (infinite) and log2diameter is typically defined only for connected graphs or for the diameter of a chosen component.

Interpreting log2diameter: it provides a logarithmic scale for the size of a graph in terms of its longest shortest path. Since Diam(G) ≥ 1 for connected graphs, log2diameter is ≥ 0, with Diam(G)=1 giving log2diameter = 0. The value is not necessarily integral, even when the diameter is integral.

Examples: A path graph on n vertices has Diam = n−1, so log2diameter = log2(n−1). A complete graph on n vertices has Diam = 1, so log2diameter = 0. A d-dimensional hypercube has Diam = d, so log2diameter = log2(d).

Computation and variants: To compute, one can run all-pairs shortest-path algorithms (such as Floyd–Warshall for weighted graphs or BFS from each node for unweighted graphs) to obtain Diam(G), then apply the logarithm. For very large graphs, exact computation may be expensive; one may use bounds based on eccentricity, radius, or approximation methods.

Applications and context: Log2diameter is a derived metric used in theoretical analyses and comparative studies where a logarithmic perspective on graph size relative to diameter is informative. It complements diameter, radius, and other centrality measures in network analysis and algorithmic complexity discussions.