Combinatorial Laplacians and Relative Homology of Complex Pairs

TL;DR

This paper extends the theory of combinatorial Laplacians to complex pairs, establishing a relative matrix-tree theorem, deriving spectral gap bounds, and linking these to relative homology, thus broadening understanding of discrete geometric structures.

Contribution

It introduces a relative matrix-tree theorem for complex pairs and explores spectral bounds and homology conditions, advancing combinatorial Laplacian theory for complex pairs.

Findings

Established a relative matrix-tree theorem for complex pairs.

Derived lower bounds for spectral gaps of complex pairs.

Provided conditions for the vanishing of relative homology.

Abstract

As a discretization of the Hodge Laplacian, the combinatorial Laplacian of simplicial complexes has garnered significant attention. In this paper, we study combinatorial Laplacians for complex pairs $(X, A)$, where $A$ is a subcomplex of a simplicial complex $X$. We establish a relative version of the matrix-tree theorem for complex pairs, which generalizes both the matrix-tree theorem for simplicial complexes proved by Duval, Klivans, and Martin (2009) and the result for Dirichlet eigenvalues of graph pairs by Chung (1996). Furthermore, we derive several lower bounds for the spectral gaps of complex pairs and characterize the equality case for one sharp lower bound. As by-products, we obtain sufficient conditions for the vanishing of relative homology. Our results demonstrate that the combinatorial Laplacians for complex pairs are closely related to relative homology.