# Condensed Ricci curvature of complete and strongly regular graphs

@article{Bonini2019CondensedRC, title={Condensed Ricci curvature of complete and strongly regular graphs}, author={Vincent Bonini and Conor Carroll and Uyen N. Dinh and Sydney Dye and Joshua Frederick and Erin P. J. Pearse}, journal={arXiv: Combinatorics}, year={2019} }

We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the Ricci curvature is strictly greater than one. We then derive explicit Ricci curvature formulas for strongly regular graphs in terms of the graph parameters and the size of a maximal matching in the core neighborhood. As a consequence we are able to derive…

## 8 Citations

### Curvatures, graph products and Ricci flatness

- MathematicsJ. Graph Theory
- 2021

It is proved that all distance-regular graphs of girth $4$ attain their maximal possible curvature values and that in the case of strong products, non-negativity of Ollivier Ricci curvature is only preserved for horizontal and vertical edges.

### Fe b 20 21 Ricci curvature , graphs and eigenvalues

- Mathematics
- 2021

We express the discrete Ricci curvature of a graph as the minimal eigenvalue of a family of matrices, one for each vertex of a graph whose entries depend on the local adjaciency structure of the…

### Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature

- Mathematics
- 2022

. An amply regular graph is a regular graph such that any two adjacent vertices have α common neighbors and any two vertices with distance 2 have β common neighbors. We prove a sharp lower bound…

### Lin-Lu-Yau curvature and diameter of amply regular graphs

- Mathematics
- 2021

By Hall’s marriage theorem, we study lower bounds of the Lin-Lu-Yau curvature of amply regular graphs with girth 3 or 4 under different parameter restrictions. As a consequence, we show that each…

### Non-negative Ollivier curvature on graphs, reverse Poincar\'e inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates

- Mathematics
- 2019

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has ‖Ptμ− Ptν‖1 ≤ W1(μ, ν) √ t for all probability measures μ, ν where Pt is the heat semigroup and W1 is the…

### Non-negative Ollivier curvature on graphs, reverse Poincaré inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates

- Mathematics
- 2019

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t \mu - P_t \nu\|_1 \leq \frac{W_1(\mu,\nu)}{\sqrt{t}} \] for all probability measures $\mu,\nu$ where…

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