  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.
