  Polterovich proved a remarkable closed formula for heat kernel coefficients
of the Laplace operator on compact Riemannian manifolds involving powers of
Laplacians acting on the distance function. In the case of K\"ahler manifolds,
we prove a combinatorial formula for powers of the complex Laplacian and use it
to derive an explicit graph theoretic formula for the numerics in heat
coefficients as a linear combination of metric jets based on Polterovich's
formula.
