Schur States, Average Mixing, and Counting Trees on Line Graphs' CTQW

We introduce a family of complex-valued edge weights on a finite simple graph $\text{G}$ arising from a continuous-time quantum walk on the line graph $\mathrm{\ell }\text{G}$, packaged as the \emph{Schur state}: an $n\times n$ Hermitian matrix encoding the amplitudes of an edge-state walk. The entrywise modulus square induces a real-weighted adjacency matrix $A(e)$ and Laplacian $L(e)$, and time-averaging yields a weighted graph whose spanning-tree count we relate to that of $\text{G}$. Our main result is $$tn\text{\hspace{0.17em}⁣}(\text{G},\frac{1}{m})=\frac{1}{m^{n-1}}tn(\text{G}),$$ valid whenever the initial edge state is \emph{uniform commutative}, where $n=\mathrm{\mid }V\text{G}\mathrm{\mid }$, $m=\mathrm{\mid }E\text{G}\mathrm{\mid }$, and $tn(\text{G},w)$ denotes the weighted spanning-tree count. We further identify a structural mechanism – the $-2$ eigenspace of $\mathrm{\ell }\text{G}$ – providing uniform commutative states beyond the regular case, in particular for line graphs of Eulerian graphs with an even number of edges. As a side result, we establish that commutative states are precisely the states whose von Neumann entropy is preserved under average mixing.