Abstract
A prominent parameter in the context of network analysis, originally proposed by Watts and Strogatz (1998), is the clustering coefficient of a graph G. It is defined as the arithmetic mean of the clustering coefficients of its vertices, where the clustering coefficient of a vertex u of G is the relative density m(G[NG(u)])∕dG(u)2 of its neighborhood if dG(u) is at least 2, and 0 otherwise. It is unknown which graphs maximize the clustering coefficient among all connected graphs of given order and size.We determine the maximum clustering coefficients among all connected regular graphs of a given order, as well as among all connected subcubic graphs of a given order. In both cases, we characterize all extremal graphs. Furthermore, we determine the maximum increase of the clustering coefficient caused by adding a single edge.
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