On graphs with exactly one anti-adjacency eigenvalue and beyond

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.