Zeta-equivalent digraphs: Simultaneous cospectrality

2014/12/15 by Peter Herbrich, Herbrich, Peter
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Spectral Theory (math.SP) #math.CO #math.SP

paper · pdf · doi:10.48550/arxiv.1412.4763

15 pages, 1 figure

arxiv created 2015/05/14 · arxiv updated 2015/05/15

Abstract

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.

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