  We present an exact analytical solution of the spectral problem of quasi
one-dimensional scaling quantum graphs. Strongly stochastic in the classical
limit, these systems are frequently employed as models of quantum chaos. We
show that despite their classical stochasticity all scaling quantum graphs are
explicitly solvable in the form $E_n=f(n)$, where $n$ is the sequence number of
the energy level of the quantum graph and $f$ is a known function, which
depends only on the physical and geometrical properties of the quantum graph.
Our method of solution motivates a new classification scheme for quantum
graphs: we show that each quantum graph can be uniquely assigned an integer $m$
reflecting its level of complexity. We show that a taut string with piecewise
constant mass density provides an experimentally realizable analogue system of
scaling quantum graphs.
