  Two graphs having the same spectrum are said to be cospectral. A pair of
singularly cospectral graphs is formed by two graphs such that the absolute
values of their nonzero eigenvalues coincide. Clearly, a pair of cospectral
graphs is also singularly cospectral but the converse may not be true. Two
graphs are almost cospectral if their nonzero eigenvalues and their
multiplicities coincide. In this paper, we present necessary and sufficient
conditions for a pair of graphs to be singularly cospectral, giving an answer
to a problem posted by Nikiforov. In addition, we construct an infinite family
of pairs of noncospectral singularly cospectral graphs with unbounded number of
vertices. It is clear that almost cospectral graphs are also singularly
cospectral but the converse is not necessarily true, we present families of
graphs where both concepts: almost cospectrality and singularly cospectrality
agree.
