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In this paper, we reformulate the Bakry-\'Emery curvature on a weighted graphin terms of the smallest eigenvalue of a rank one perturbation of the so-calledcurvature matrix using Schur complement. This new viewpoint allows us to showvarious curvature function properties in a very conceptual way. We show thatthe curvature, as a function of the dimension parameter, is analytic, strictlymonotone increasing and strictly concave until a certain threshold after whichthe function is constant. Furthermore, we derive the curvature of the Cartesianproduct using the crucial observation that the curvature matrix of the productis the direct sum of each component. Our approach of the curvature functions ofgraphs can be employed to establish analogous results for the curvaturefunctions of weighted Riemannian manifolds. Moreover, as an application, weconfirm a conjecture (in a general weighted case) of the fact that thecurvature does not decrease under certain graph modifications.

Bakry-\'Emery curvature on graphs as an eigenvalue problem