The anti-regular graph conjecture on extremal eigenvalues of threshold graphs

A threshold graph is a graph obtained from an isolated vertex by repeatedly adding either an isolated vertex or a dominating vertex. Let $A_n$ denote the anti-regular graph on $n$ vertices, namely the threshold graph with only two vertices of equal degree. Anti-regular graph conjecture. For each $n$, among all threshold graphs on $n$ vertices, $A_n$ has the smallest positive eigenvalue and the largest negative eigenvalue less than $-1$. This conjecture concerns the extremal nontrivial eigenvalues of threshold graphs. Earlier work established partial results and identified critical cases requiring a more refined method; the conjecture is presented here as the problem addressed by the paper.