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The Ollivier curvature has important applications in discrete geometry andnetwork theory, in particular as a measure of local clustering. The Olliviercurvature is defined in terms of the Wasserstein distance which, in thediscrete setting, can be regarded as an optimal solution of a particular linearprogramme. In certain classes of graph, this linear programme may be solved\textit{a priori} giving rise to exact combinatorial expressions for theOllivier curvature. It has been claimed by Bhattacharya and Mukherjee (2013)that an exact expression exists for the Ollivier curvature in bipartite graphsand graphs of girth 5; we present counterexamples to these claims and identifythe error in the argument of Bhattacharya and Mukherjee. We then repeat theanalysis of Bhattacharya and Mukherjee for arbitrary graphs, taking this errorinto account, and present reduced---parallelly solvable---linear programmes forthe calculation of the Ollivier curvature. This allows for potentialimprovements in the exact numerical evaluation of the Ollivier curvature,though the result heuristically suggests no general exact combinatorialexpression for the Ollivier curvature exists. Finally we give an exactexpression for the Ollivier curvature in a class of graphs defined by aparticular combinatorial constraint motivated by physical considerations.

Exact Expressions and Reduced Linear Programmes for the Ollivier Curvature in Graphs