On the inverse of unicyclic 3-coloured digraphs

Abstract

This article provides a combinatorial description of the inverse of the adjacency matrix of a non-singular 3-coloured digraph. The class of unicyclic 3-coloured digraphs with the cycle weight and with a unique perfect matching, denoted by , is considered in this article. We characterize the 3-coloured digraphs in whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs in whose inverses are bipartite are also characterized. It is proved that the inverses of the 3-coloured digraphs in are always Laplacian non-singular. Characterizations of unicyclic 3-coloured digraphs in possessing unicyclic inverses are also supplied in this article. As an application, we can obtain the class of unicyclic 3-coloured digraphs with the cycle weight satisfying the strong reciprocal eigenvalue property.