Analytic and Reidemeister torsions of digraphs and path complexes

Abstract

We define the notions of Reidemeister torsion and analytic torsion for directed graphs by means of the path homology theory introduced by the authors in [ $\href{https://arxiv.org/abs/1207.2834}{7}$, $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3324763}{8}$, $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3431683}{9}$, $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3845076}{11}$]. We prove the identity of the two notions of torsions as well as obtain formulas for torsions of Cartesian products and joins of digraphs.