Kernels of Directed Graph Laplacians

Abstract. Let G denote a directed graph with adjacency matrix Q and indegree matrix D. We consider the Kirchhoff matrix L = D - Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graphtheoretic property determines the dimension of this eigenspace

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: John S. Caughman IV, J. J. P. Veerman
• Keywords: COMBINATORICS 2006, Directed Graphs, Indegree Matrix D, Matrix