On singularity and properties of eigenvectors of complex Laplacian matrix of multidigraphs

Abstract

AbstractIn this article, we associate a Hermitian matrix to a multidigraph G. We call it the complex Laplacian matrix of G and denote it by [Formula: see text]. It is shown that the complex Laplacian matrix is a generalization of the Laplacian matrix of a graph. But, unlike the Laplacian matrix of a graph, the complex Laplacian matrix of a multidigraph may not always be singular. We obtain a necessary and sufficient condition for the complex Laplacian matrix of a multidigraph to be singular. For a multidigraph G, if [Formula: see text] is singular, we say G is [Formula: see text]-singular. We generalize some properties of the Fiedler vectors of undirected graphs to the eigenvectors corresponding to the second smallest eigenvalue of [Formula: see text]-singular multidigraphs.

Keywords

• Multidigraph
• complex adjacency matrix
• complex Laplacian matrix
• spectrum
• Fiedler vector
• 05C50