Normalized hodge laplacian matrix and its spectrum

Abstract

Let 𝑋 be a simplicial complex, 𝜕[ Subscript 𝑘] ∶ 𝐶[ Subscript 𝑘] → 𝐶[ Subscript 𝑘−1] a boundary map on 𝑋 and 𝐵[ Subscript 𝑘] a matrix representation of 𝜕[ Subscript 𝑘]. A Hodge 𝑘-Laplacian matrix on simplicial complexes is defined by 𝐿[ Subscript 𝑘]= 𝐵𝑘+1𝐵𝑇 𝑘+1 + 𝐵𝑇 𝑘 𝐵𝑘 which is a generalization of a Laplacian matrix 𝐿 on graphs. In this work, we study a Hodge 𝑘-Laplacian matrix and then generalize a normalized Laplacian matrix ℒ on graphs to a normalized Hodge 𝑘- Laplacian matrix ℒ[ Subscript 𝑘] (i.e. ℒ[ Subscript 0]= ℒ) on simplicial complexes. This matrix is also a Hodge Laplacian matrix and this fact leads some useful properties. We also obtain that the smallest eigenvalue of a normalized Hodge 𝑘-Laplacian indicates whether the homology (or cohomology) on a given simplicial complex is trivial. We demonstrate eigenvalues of ℒ[ Subscript 𝑘] for some special cases and study a relation between its eigenvalues and 𝑞-wedge sum of simplices. We finally apply this matrix for random walks on simplicial complexes.