Extremal graphs for the $k$-th eigenvalue

Abstract

For a simple graph $G$ of order $n$, let $λ_1(G)\ge \cdots \ge λ_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $λ_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ λ_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp examples arising from maximal real equiangular tight frames. In this paper, we characterize the equality case. We also obtain an explicit combinatorial description of the extremal graphs for $λ_3$ and $λ_4$.