Kernels of Directed Graph Laplacians

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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
John S. Caughman IV, J. J. P. Veerman
Added 11 Dec 2010
Updated 11 Dec 2010
Type Journal
Year 2006
Authors John S. Caughman IV, J. J. P. Veerman
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