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

Abstract

In this article, we associate a Hermitian matrix to a multidigraph G. We call it the complex Laplacian matrix of G and denote it by $L_{\mathrm{}\mathbb{C}}(G)$. 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 $L_{\mathrm{}\mathbb{C}}(G)$ is singular, we say G is $L_{\mathrm{}\mathbb{C}}$-singular. We generalize some properties of the Fiedler vectors of undirected graphs to the eigenvectors corresponding to the second smallest eigenvalue of $L_{\mathrm{}\mathbb{C}}$-singular multidigraphs.

KEYWORDS:

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

AMS SUBJECT CLASSIFICATION:

• 05C50
• 05C05
• 15A18

Acknowledgments

The authors are thankful to the anonymous referees for a careful reading of the article and the encouraging comments made in the report.

Notes

2 By a multi-directed edge between two vertices i and j in a multidigraph, we mean all the directed edges between i and j.

Additional information

Funding

The corresponding author acknowledges SERB, Government of India, for financial support through grant (MTR/2017/000080). The second author acknowledges the financial support from UGC, Government of India.