Morse complexes and <i>M</i> -torsions of digraphs

In this paper, we define Reidemeister torsion and analytic torsion for digraphs by means of Morse theory (abbreviated as M-torsion) of digraphs. We study the structure of Morse complex of digraphs and join of digraphs, illustrate through some typical examples that M-torsions are different from torsions defined by the path complex, and prove that M-torsion of contractible digraphs and cone are all equal to 1.