Global Poincar\'e inequality on Graphs via Conical Curvature-Dimension Conditions

Sajjad Lakzian; Zachary McGuirk

Global Poincar\'e inequality on Graphs via Conical Curvature-Dimension Conditions

Differential Geometry 2018-07-26 v2

Authors: Sajjad Lakzian , Zachary McGuirk

Abstract

We introduce and study the conical curvature-dimension condition, $CCD(K,N)$ , for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincar\'e inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the conical curvature-dimension analysis is finding a sharp estimate on the curvature of complete graphs.

Keywords

conical curvature-dimension condition global Poincaré inequality graphs first eigenvalue sharp lower bound complete graphs curvature estimate

Cite

@article{arxiv.1605.05432,
  title  = {Global Poincar\'e inequality on Graphs via Conical Curvature-Dimension Conditions},
  author = {Sajjad Lakzian and Zachary McGuirk},
  journal= {arXiv preprint arXiv:1605.05432},
  year   = {2018}
}

Comments

15 pages, minor corrections in v2 and generalized harmonic functions are getting their own manuscript so they have been removed from this version

Global Poincar\'e inequality on Graphs via Conical Curvature-Dimension Conditions

Abstract

Keywords

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