Zeta-equivalent digraphs: Simultaneous cospectrality

We introduce a zeta function of digraphs that determines, and is determined by, the spectra of all linear combinations of the adjacency matrix, its transpose, the out-degree matrix, and the in-degree matrix. In particular, zeta-equivalence of graphs encompasses simultaneous cospectrality with respect to the adjacency, the Laplacian, the signless Laplacian, and the normalized Laplacian matrix, respectively. In addition, we express zeta-equivalence in terms of Markov chains and in terms of invasions where each edge is replaced by a fixed digraph. We finish with a method for constructing zeta-equivalent digraphs.