Graphs with nonnegative Bakry-\'Emery curvature without Quadrilateral

The definition of Ricci curvature on graphs in Bakry-\'Emery's sense based on curvature dimension condition was introduced by Lin and Yau Math. Res. Lett. , 2010. Hua and Lin Comm. Anal. Geom. , 2019 classified unweighted graphs satisfying the curvature dimension condition CD (0, ) whose girth are at least five. In this paper, we classify all of connected unweighted normalized C₄-free graphs satisfying curvature dimension condition CD (0, ) for minimum degree at least 2 and the case with non-normalized Laplacian without degree condition. .