From Wikipedia:
"If there is no path connecting two
vertices, i.e., if they belong to
different connected components, then
conventionally the distance is defined
as infinite."
This seems to negate the possibility that there are graphs with vertices connected by an infinite shortest path (as opposed to being not connected).
Why is it that for every (even infinite) path between two vertices there is a finite one?
Note that infinite paths between vertices do exist – e.g. in the infinite complete graph -, but they are not the shortest.
Best Answer
To expand on my comment: It's clear that if an infinite path is defined as a map from $\mathbb N$ to the edge set such that consecutive edges share a vertex, then any vertices connected by such an infinite path are in fact connected by a finite section of the path. To make sense of the question nevertheless, one might ask whether it is possible to use a different ordinal than $\omega$, say, $\omega\cdot2$, to define an infinite path. But that doesn't make sense either, since there's no way (at least I don't see one) to make the two parts of such a path have anything to do with each other -- at each limit ordinal, the path can start wherever it wants, since there's no predecessor for applying the condition that consecutive edges share a vertex.
Note that the situation is different in infinite trees, which can perfectly well contain infinite paths connecting the root to a node. This is because the definition of a path in an infinite tree is different; it explicitly attaches the nodes on levels corresponding to limit ordinals to entire sequences of nodes, not to individual nodes; such a concept doesn't exist in graphs.