Graphs with positive Lin-Lu-Yau curvature without quadrilaterals
Huiqiu Lin, Zhe You
TL;DR
The paper classifies all simple connected $C_4$-free graphs with minimum degree at least $2$ that have positive Lin-Lu-Yau curvature, showing the only possibilities are $C_3$, $C_5$, the graphs $T$ and $H$ (line graph of the Petersen graph), and the friendship graphs $F_2$ and $F_3$. Using the limit-free Laplacian formulation of $\kappa_{\mathrm{LLY}}$, it derives structural constraints (notably $\Delta(G)\le6$) and performs case analysis to obtain this finite list. It verifies positivity for the candidate graphs (some by explicit estimates, others via a curvature calculator). The work further discusses extensions to graphs with pendant edges and highlights limitations under induced $C_4$-free relaxations, proposing future directions for classifying graphs with non-negative curvature.
Abstract
The definition of Ricci curvature on graphs was given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Math., 2009. Recently, a powerful limit-free formulation of Lin-Lu-Yau curvature using the graph Laplacian has been given in Münch-Wojciechowski, Adv. Math., 2019. Let $F_k$ be the friendship graph obtained from $k$ triangles by sharing a common vertex and $T$ be the graph obtained from a triangle and $K_{1,3}$ by adding a matching between every leaf of $K_{1,3}$ and a vertex of the triangle. In this paper, we classify all the simple connected $C_4$-free graphs with positive Lin-Lu-Yau curvature for minimum degree at least 2: the cycles $C_3,C_5$, the friendship graphs $F_2,F_3$, the line graph of Peterson graph, and $T$.