  In a recent article J. Phys. Compl. 4 (2023) 035005, Kawamoto evoked
statistical physics methods for the problem of counting graphs with a
prescribed degree sequence. This treatment involved truncating a particular
Taylor expansion at the first two terms, which resulted in the Bender-Canfield
estimate for the graph counts. This is surprisingly successful since the
Bender-Canfield formula is asymptotically accurate for large graphs, while the
series truncation does not a priori suggest a similar level of accuracy.
  We upgrade the above treatment in three directions. First, we derive an exact
formula for counting d-regular graphs in terms of a d-dimensional Gaussian
integral. Second, we show how to convert this formula into an integral
representation for the generating function of d-regular graph counts. Third, we
perform explicit saddle point analysis for large graph sizes and identify the
saddle point configurations responsible for graph count estimates. In these
saddle point configurations, only two of the integration variables condense to
significant values, while the remaining ones approach zero for large graphs.
This provides an underlying picture that justifies Kawamoto's earlier findings.
