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, applying the definition of edge
cuts as cuts that increase the number of connected component gives 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.
