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Curvatures, graph products and Ricci flatness

Tutkimustuotosvertaisarvioitu

Yksityiskohdat

AlkuperäiskieliEnglanti
JulkaisuJOURNAL OF GRAPH THEORY
TilaSubmitted - 25 syyskuuta 2019
OKM-julkaisutyyppiA1 Alkuperäisartikkeli

Tiivistelmä

In this paper, we compare Ollivier Ricci curvature and Bakry-\'Emery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-\'Emery curvature under triangle-freeness and an additional in-degree condition. We also provide examples that both conditions of this result are necessary. We investigate relations to graph products and show that Ricci flatness is preserved under all natural products. While non-negativity of both curvatures are preserved under Cartesian products, we show that in the case of strong products, non-negativity of Ollivier Ricci curvature is only preserved for horizontal and vertical edges. We also prove that all distance-regular graphs of girth $4$ attain their maximal possible curvature values.

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