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The S-metric has grown popular in network studies, as a measure of``scale-freeness'' restricted to the collection G(D) of connected graphs with acommon degree sequence D=(d_1,\ldots,d_n). The calculation of S depends on themaximum possible degree assortativity r among graphs in G(D). The originalmethod involves a heuristic construction of a maximally assortative graph g*.The approximation by Beichl and Cloteaux involves constructing a possiblydisconnected graph g' with r(g') >= r(g*) and requires O(n^2) tests for thegraphicality of a degree sequence. The present paper uses the Tripathi-Vijaytest to streamline this approximation, and thereby to investigate twocollections of graphs: Barabasi-Albert trees and coauthorship graphs ofmathematical sciences researchers. Long-term trends in the coauthorship graphsare discussed, and contextualized by insights derived from the BA trees. It isknown that greater degree-based preferential attachment produces greatervariance in degree sequences, and these trees exhibited assortativitiesrestricted to a narrow band. In contrast, variance in degree rose over time inthe coauthorship graphs in spite of weakening degree-based preferentialattachment. These observations and their implications are discussed and avenuesof future work are suggested.

The S-metric, the Beichl-Cloteaux approximation, and preferential attachment