Quartic graphs which are Bakry-Émery curvature sharp

Abstract

We give a classification of all connected quartic graphs which are (infinity) curvature sharp in all vertices with respect to Bakry-Émery curvature. The result is based on a computer classification by F. Gurr and L. Watson May and a combinatorial case by case investigation. 2 CUSHING, KAMTUE, PEYERIMHOFF, AND WATSON MAY equality. The main result in the paper is a complete classification of all 4-regular (quartic) curvature sharp graphs: Theorem 1.1. Let G = (V, E) be a connected quartic graph which is Bakry-Émery curvature sharp in all vertices. Then G is one of the following: (i) The complete graph K 5 with |V | = 5, K ∞ = 3.5, diam G = 1; (ii) The octahedral graph O with |V | = 6, K ∞ = 3, diam G = 2; (iii) The Cartesian product K 3 × K 3 of two copies of the complete graph K 3 with |V | = 9, K ∞ = 2.5, diam G = 2; (iv) The complete bipartite graph K 4,4 with |V | = 8, K ∞ = 2, diam G = 2; (v) The crown graph C(10) with |V | = 10, K ∞ = 2, diam G = 3; (vi) The Cayley graph Cay(D 12 , S) of the dihedral group D 12 of order 12 with generators S = {r 3 , s, sr 2 , sr 4 } with |V | = 12, K ∞ = 2, diam G = 3; (vii) The Cayley graph Cay(D 14 , S) of the dihedral group D 14 of order 14 with generators S = {s, sr, sr 4 , sr 6 } with |V | = 14, K ∞ = 2, diam G = 3; (viii) The 4-dimensional hypercube Q 4 with |V | = 16, K ∞ = 2, diam G = 4.

Bakry-Émery curvature on graphs as an eigenvalue problem

“…In recent years, the discrete Bakry-Émery theory on graphs has become an active emerging research field. There are a growing number of articles investigating this theory, see e.g., [7,8,9,10,11,12,14,15,17,20,21,22,23,24,26,27,28,29,31,32,33,34,35,37,38,39,41,42,44,45,47,48,49,52]. Let us mention here important related works on non-linear discrete curvature dimension inequalities, see e.g., [4,13,18,19,43].…”

“…The upper bound, in particular, plays an important role in our curvature analysis, where we study the situation when this upper bound is attained (called curvature sharpness; see the definition below). The notion of curvature sharpness was introduced [12] and studied in, e.g., [10].…”