SpanningvalminG

SpanningvalminG is a notational term used in graph theory to denote the minimum spanning value of a graph G with respect to a specified edge valuation. Given a graph G = (V,E) and a weight function w: E -> R assigning a numeric value to each edge, spanningvalminG is defined as the minimum possible total weight among all spanning trees of G: spanningvalminG = min { sum_{e in T} w(e) : T is a spanning tree of G }. If G is connected, the minimum is always achieved by a minimum spanning tree (MST). If G is not connected, the concept is typically extended to a minimum spanning forest or is considered undefined, depending on the context.

In weighted, undirected graphs with nonnegative weights, spanningvalminG equals the total weight of an MST of G. The value is invariant under different MSTs when multiple exist, though they may differ in edge composition while sharing the same total weight. Algorithms to compute spanningvalminG include Kruskal's algorithm and Prim's algorithm, with time complexities commonly cited as O(E log V) and O(E + V log V) respectively, using data structures such as disjoint-set unions.

Beyond undirected graphs, analogous notions exist for directed graphs, yielding minimum arborescences (via Edmonds' algorithm). SpanningvalminG is a fundamental quantity in network design, clustering, and reliability, serving as a standard measure of the most cost-efficient way to connect all vertices with a spanning structure.

See also: minimum spanning tree, Kruskal's algorithm, Prim's algorithm, minimum spanning forest, Edmonds' algorithm.