Graphs with nonnegative Bakry-Émery curvature without quadrilateral

The definition of Ricci curvature on graphs in Bakry-Émery’s sense based on curvature dimension condition was introduced by Lin and Yau [Math. Res. Lett. 17 (2010), pp. 343–356]. Hua and Lin [Comm. Anal. Geom. 27 (2019), pp. 619–638] classified unweighted graphs satisfying the curvature dimension condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper D left-parenthesis 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">CD(0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose girth are at least five. In this paper, we classify all of connected unweighted normalized <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 4"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">C_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -free graphs satisfying curvature dimension condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper D left-parenthesis 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">CD(0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for minimum degree at least 2 and the case with non-normalized Laplacian without degree condition.