Kernels of Directed Graph Laplacians

Abstract

Let $G$ denote a directed graph with adjacency matrix $Q$ and in-degree 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 recently 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 graph-theoretic property determines the dimension of this eigenspace – namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights.

Entropic effects in a nonequilibrium system: Flocks of birds

“…Since G has only one connected component, there is a single zero eigenvalue of Λ [29], which we denote by λ 1 , and Eq. (27) shows that the probability p mot (G) is determined by the product of the nonzero eigenvalues of the Laplacian (19).…”

On eigenvalues of Laplacian matrix for a class of directed signed graphs

“…Proof. By taking into account Remark 1, the fact that the eigenvalue is semisimple and the results regarding γ k correspond to [11,Corollary 4.1]. Denote A the adjacency matrix of G. By relabeling the nodes, they can be partitioned such that the first X 1 nodes in the first partition belong to the set X 1 , the second X 2 nodes in the second partition belong to the set X 2 , and so on, and the remaining nodes belong to the set C = d ⋃ k=1 C k .…”

Sparse Control for Dynamic Movement Primitives

“…Remark 6. The conditions of a bidirectional coupling K ij = K ji can be relaxed to consider directional couplings in the case that K ij = K = K T > 0 for a fixed K. The sparse inhibition result holds if all nodes are reachable from the inhibited node (Caughman and Veerman, 2006).…”