Complex adjacency spectra of digraphs

Abstract

In this article, we consider only those (simple) digraphs which satisfy the property that if $(u,v)$ is an edge of a digraph, then $(v,u)$ is not an edge of it. A new matrix representation of a digraph is considered and the matrix is named as the complex adjacency matrix. The eigenvalues and the eigenvectors of the complex adjacency matrices of cycle digraphs and directed trees are obtained and it is shown that not only the eigenvalues of these matrices but also the eigenvectors provide a lot of information about the structure of these digraphs.

COMMUNICATED BY:

• B. Shader

Keywords:

• Digraphs
• complex adjacency matrix
• complex adjacency spectrum
• cospectral co-base digraphs
• counter spectral co-base digraphs

Acknowledgements

The author is thankful to the referee for his/her valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The author acknowledges the financial support from the University Grants Commission (UGC), Government of India [F. 2-1/2013(SA-1)] and Indian Statistical Institute, Delhi Centre.