  Continuing previous work we develop a certain piece of functional analysis on
general graphs and use it to create what Connes calls a 'spectral triple', i.e.
a Hilbert space structure, a representation of a certain (function) algebra and
a socalled 'Dirac operator', encoding part of the geometric/algebraic
properties of the graph. We derive in particular an explicit expression for the
'Connes-distance function' and show that it is in general bounded from above by
the ordinary distance on graphs (being, typically, strictly smaller(!) than the
latter). We exhibit, among other things, the underlying reason for this
phenomenon.
