  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 a
common degree sequence D=(d_1,\ldots,d_n). The calculation of S depends on the
maximum possible degree assortativity r among graphs in G(D). The original
method involves a heuristic construction of a maximally assortative graph g*.
The approximation by Beichl and Cloteaux involves constructing a possibly
disconnected graph g' with r(g') >= r(g*) and requires O(n^2) tests for the
graphicality of a degree sequence. The present paper uses the Tripathi-Vijay
test to streamline this approximation, and thereby to investigate two
collections of graphs: Barabasi-Albert trees and coauthorship graphs of
mathematical sciences researchers. Long-term trends in the coauthorship graphs
are discussed, and contextualized by insights derived from the BA trees. It is
known that greater degree-based preferential attachment produces greater
variance in degree sequences, and these trees exhibited assortativities
restricted to a narrow band. In contrast, variance in degree rose over time in
the coauthorship graphs in spite of weakening degree-based preferential
attachment. These observations and their implications are discussed and avenues
of future work are suggested.
