  The principal graph $X$ of a subfactor with finite Jones index is one of the
important algebraic invariants of the subfactor. If $\Delta$ is the adjacency
matrix of $X$ we consider the equation $\Delta=U+U^{-1}$. When $X$ has square
norm $\leq 4$ the spectral measure of $U$ can be averaged by using the map
$u\to u^{-1}$, and we get a probability measure $\epsilon$ on the unit circle
which does not depend on $U$. We find explicit formulae for this measure
$\epsilon$ for the principal graphs of subfactors with index $\le 4$, the
(extended) Coxeter-Dynkin graphs of type $A$, $D$ and $E$. The moment
generating function of $\epsilon$ is closely related to Jones' $\Theta$-series.
