  We study a family of closed quantum graphs described by one singular vertex
of order n=4. By suitable choice of the parameters specifying the singular
vertex, we can construct a closed sequence of paths in the parameter space that
physically corresponds to the smooth interpolation of different topologies - a
ring, separate two lines, separate two rings, two rings with a contact point.
We find that the spectrum of a quantum particle on this family of graphs shows
the quantum anholonomy.
