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arXiv · 2607.04878

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

Kevin Fung·Johnny Lim
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Abstract

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.

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Kevin Fung, Johnny Lim. 2026-07-06. Steinerberger Curvature On Digraphs -- Discrete Bonnet-Myers and Lichnerowicz Theorems. https://arxiv.org/abs/2607.04878

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