Given any graph, we can uniquely associate a square matrix which stores informations about its vertices and how they are interconnected. The goal of spectral graph theory is to see how the eigenvalues and eigenvectors of such a matrix representation of a graph are related to the graph structure. We consider here (multi)digraphs and define a new matrix representation for a multidigraph and named it as the complex adjacency matrix.

The relationship between the adjacency matrix and the complex adjacency matrix of a multidigraph are established. Furthermore, some of the advantages of the complex adjacency matrix over the adjacency matrix of a multidigraph are observed. Besides, some of the interesting spectral properties (with respect to the complex adjacency spectra) of a multidigraph are established. It is shown that not only the eigenvalues, but also the eigenvectors corresponding to the complex adjacency matrix of a multidigraph carry a lot of information about the structure of the multidigraph.

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Last updated: 20 Mar 2023