nLab connected graph

Idea

An inhabited graph has two notions of being connected:

• weak connected - it cannot be decomposed as the disjoint union of two inhabited graphs.

• strong or path connected - each pair of vertices is connected by a sequence of edges or in other words every homset of its free category is inhabited.

If a graph’s edges are undirected these two notions coincide while if directed the stronger condition implies the weaker.

For the various categories of graphs, which of their objects are weakly connected correspond to the general notion of connected objects in these categories.

Examples

• In perturbative quantum field theory the formal power series of Feynman amplitudes labeled by those Feynman diagrams whose underlying graphs are connected is called the effective action $S_{eff}$. The exponential of the effective action then yields the full Feynman perturbation series over all Feynman diagrams. See at S-matrix for more.

References

• Wikipedia, Connectivity (graph theory)