Steinerberger Curvature On Digraphs -- Discrete Bonnet-Myers and Lichnerowicz Theorems

Steinerberger curvature encodes the global distance geometry of a graph through an equilibrium measure. In this paper, we derive explicit curvature formulas for undirected Cayley graphs of dihedral groups $D_n$ and generalized quaternion groups $Q_{4m}$. We then extend Steinerberger curvature to strongly connected simple digraphs by introducing in-curvature and out-curvature, reflecting the asymmetry of directed distances. For these directed curvatures, we establish structural properties, including negativity criteria and a permutation relation between in- and out-curvature. Our main results are directed analogues of the Bonnet–Myers, Cheng and Lichnerowicz theorems, together with reverse Bonnet–Myers inequalities for directed diameter and out-radius, and an upper bound for in-radius in terms of total curvature.