  The anti-adjacency matrix of a graph 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 in 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 eigenvalue, which is
an analog of Smith's classical result that a connected graph with exactly one
positive adjacency eigenvalue iff it is a complete multipartite graph. On this
basis, we identify the connected graphs with all but at most two anti-adjacency
eigenvalues equal to $-2$ and $0$. Moreover, for the anti-adjacency matrix we
determine the HL-index of graphs with exactly one positive anti-adjacency
eigenvalue, where the HL-index measures how large in absolute value may be the
median eigenvalues of a graph. We finally propose some problems for further
study.
