Large Values of the Clustering Coefficient

TL;DR

This paper investigates the maximum possible clustering coefficient in various classes of graphs, characterizes the extremal graphs achieving these maxima, and examines how adding an edge influences the clustering coefficient.

Contribution

It provides exact maximum clustering coefficients for connected regular and subcubic graphs and characterizes all extremal graphs, also analyzing the impact of adding a single edge.

Findings

Identifies maximum clustering coefficients for connected regular graphs.

Identifies maximum clustering coefficients for connected subcubic graphs.

Quantifies the increase in clustering coefficient from adding one edge.

Abstract

A prominent parameter in the context of network analysis, originally proposed by Watts and Strogatz (Collective dynamics of `small-world' networks, Nature 393 (1998) 440-442), 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 $m(G[N_G(u)])/{d_G(u)\choose 2}$ of its neighborhood if $d_G(u)$ 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.