  The alignment strength of a graph matching is a quantity that gives the
practitioner a measure of the correlation of the two graphs, and it can also
give the practitioner a sense for whether the graph matching algorithm found
the true matching. Unfortunately, when a graph matching algorithm fails to find
the truth because of weak signal, there may be "phantom alignment strength"
from meaningless matchings that, by random noise, have fewer disagreements than
average (sometimes substantially fewer); this alignment strength may give the
misleading appearance of significance. A practitioner needs to know what level
of alignment strength may be phantom alignment strength and what level
indicates that the graph matching algorithm obtained the true matching and is a
meaningful measure of the graph correlation. The {\it Phantom Alignment
Strength Conjecture} introduced here provides a principled and practical means
to approach this issue. We provide empirical evidence for the conjecture, and
explore its consequences.
