A note on Steinerberger's curvature for graphs

TL;DR

This paper explores Steinerberger curvature formulas for block graphs, examines curvature relations between graphs, and characterizes self-centered Bonnet-Myers sharp graphs as antipodal, comparing Steinerberger and Ollivier Ricci curvature results.

Contribution

It introduces Steinerberger curvature formulas for block graphs and characterizes certain sharp graphs, providing new insights into graph curvature relations and properties.

Findings

Steinerberger curvature formulas derived for block graphs

Curvature relations between graphs connected by a bridge analyzed

Self-centered Bonnet-Myers sharp graphs characterized as antipodal

Abstract

In this note, we provide Steinerberger curvature formulas for block graphs, discuss curvature relations between two graphs and the graph obtained by connecting them via a bridge, and show that self-centered Bonnet-Myers sharp graphs are precisely those which are antipodal. We also discuss similarities and differences between Steinerberger and Ollivier Ricci curvature results.