  We analyze the heat kernel associated to the Laplacian on a compact metric
graph, with standard Kirchoff-Neumann vertex conditions. An explicit formula
for the heat kernel as a sum over loops, developed by Roth and Kostrykin,
Potthoff, and Schrader, allows for a straightforward analysis of small-time
asymptotics. We show that the restriction of the heat kernel to the diagonal
satisfies a modified version of the heat equation. This observation leads to an
"edge" heat trace formula, expressing the a sum over eigenfunction amplitudes
on a single edge as a sum over closed loops containing that edge. The proof of
this formula relies on a modified heat equation satisfied by the diagonal
restriction of the heat kernel. Further study of this equation leads to
explicit formulas for completely symmetric graphs.
