We study the heat kernel expansion of the Laplacian on n-forms defined on a subgraph of a directed complete graph. We derive two expressions for the subgraph heat kernel on 0-forms and compute the coefficients of the expansion. We also obtain the subgraph heat kernel of the Laplacian on 1-forms.

There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect function of the Vaidya spacetime.

For Laplacians defined by measures on a bounded domain in R^n, we prove analogs of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Polya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.