  The anti-adjacency matrix of graphs is constructed from the distance matrix
of a graph by keeping each row and each column only the largest distances. This
matrix can be interpreted as the opposite of the adjacency matrix, which is
instead constructed from the distance matrix of a graph by keeping each row and
each column only the distances equal to 1. The anti-adjacency eigenvalues of a
graph are those of its anti-adjacency matrix. Employing a novel technique
introduced by Haemers [Spectral characterization of mixed extensions of small
graphs, Discrete Math. 342 (2019) 2760--2764], we characterize all connected
graphs with exactly one positive anti-adjacency eigenvalues. On this basis, we
identify the connected graphs with all but at most two anti-adjacency
eigenvalues equal to -2 and 0. We finally propose some problems for further
study.
