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We consider particular weighted directed graphs with edges having colour red, blue or green such that each red edge has weight 1, each blue edge has weight $ -1 $ − 1 and each green edge has weight $ \mathrm {i} $ i (the imaginary unit). Such a directed graph is called a 3-coloured digraph (for short, a 3-CD). Every mixed graph and every signed graph can be interpreted as a 3-CD. We first study some structural properties of 3-CDs by means of the eigenvalues of the adjacency matrix. In particular, we give spectral criteria for singularity of such digraphs. Second, we consider weight-symmetric 3-CDs, i.e. those 3-CDs that are switching isomorphic to their negation. It follows that the class of weight-symmetric 3-CDs is included in the class of 3-CDs whose spectrum is symmetric (with respect to the origin). We give some basic properties and several constructions of weight-symmetric 3-CDs and establish constructions of 3-CDs which have symmetric spectrum but are not weight-symmetric. [ABSTRACT FROM AUTHOR]

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 ±i and with a unique perfect matching, denoted by U g , is considered in this article. We characterize the 3-coloured digraphs in U g whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs in U g whose inverses are bipartite are also characterized. It is proved that the inverses of the 3-coloured digraphs in U g are always Laplacian non-singular. Characterizations of unicyclic 3-coloured digraphs in U g 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 ±i satisfying the strong reciprocal eigenvalue property. [ABSTRACT FROM AUTHOR]