  Berry and Wang [Phys. Rev. A {\bf 83}, 042317 (2011)] show numerically that a
discrete-time quantum random walk of two noninteracting particles is able to
distinguish some non-isomorphic strongly regular graphs from the same family.
Here we analytically demonstrate how it is possible for these walks to
distinguish such graphs, while continuous-time quantum walks of two
noninteracting particles cannot. We show analytically and numerically that even
single-particle discrete-time quantum random walks can distinguish some
strongly regular graphs, though not as many as two-particle noninteracting
discrete-time walks. Additionally, we demonstrate how, given the same quantum
random walk, subtle differences in the graph certificate construction algorithm
can nontrivially impact the walk's distinguishing power. We also show that no
continuous-time walk of a fixed number of particles can distinguish all
strongly regular graphs when used in conjunction with any of the graph
certificates we consider. We extend this constraint to discrete-time walks of
fixed numbers of noninteracting particles for one kind of graph certificate; it
remains an open question as to whether or not this constraint applies to the
other graph certificates we consider.
