  In this paper, we generalize the combinatorial Laplace operator of Horak and
Jost by introducing the $\phi$-weighted coboundary operator induced by a weight
function $\phi$. Our weight function $\phi$ is a generalization of Dawson's
weighted boundary map. We show that our above-mentioned generalizations include
new cases that are not covered by previous literature. Our definition of
weighted Laplacian for weighted simplicial complexes is also applicable to
weighted/unweighted graphs and digraphs.
