There are several definitions of the strength of a graph.
Harary and Palmer (1959) and Harary and Palmer (1973, p. 66) define the strength of a tree as the maximum number of edges between any pair of vertices. This definition corresponds to a tree's graph diameter.
Harary and Palmer (1973, p. 117) define the strength of a multigraph as the maximum number of edges joining any two adjacent vertices.
Capobianco and Molluzzo (1979-1980) define the strength of a separable graph as , where the strength vector of a graph is defined as the vector of increases in the connected component count upon deletion of vertex . For example, the Capobianco-Molluzzo strength vector of the graph illustrated above is . The Capobianco-Molluzzo strength of a nonseparable graph is then defined to be .
The most standard definition of the strength of a simple connected graph is
where is the number of connected components and the minimum is taken over all edge cuts of (Gusfield 1983, 1991). Here, the subtraction by one in the denominator gives the number of additional connected components created. Graph strength therefore gives a measure of the resistance of a graph to edge-deletion, and so is a measure of vulnerability of a network to attack (Cunningham 1985, Gusfield 1991) and can be naturally generalized to edge-weighted graphs. Computing the strength of a graph can be done in polynomial time (Cunningham 1985, Trubin 1993).
While one could take for disconnected graphs as done by Chvátal (1973) in the analogous case of graph toughness, using the definition of edge cuts as cuts that increase the number of connected components, the definition can be applied to give well-defined strengths for disconnected graphs.
A vertex cut analog of graph strength is known as graph toughness.
The Tutte-Nash-Williams theorem states that , where is the floor function, is the maximum number of edge-disjoint spanning trees that can be contained in a graph (Gusfield 1984, Cunningham 1985).
Graph strength is unrelated to the graph strong product.