The Sharp Upper Bounds for the Median Eigenvalues of Graphs

Abstract

Let $λ_1\geqλ_2\geq\cdots\geqλ_n$ be the eigenvalues of a simple graph $G$ of order $n$. The HL-index of $G$ is defined by $R(G)=\max\|λ_h|,|λ_\ell|\}$ with $h=\lfloor(n+1)/2\rfloor$ and $\ell=\lceil(n+1)/2\rceil$.In this paper, we prove that if $G$ is $ K_4$-minor-free or $ K _ {2,3} $-minor-free, then $R(G)\leq\sqrt{5}-1$ with equality attained by an infinite family of outerplanar graphs.Moreover, we show that $R(G)\leq\sqrt{d-2}$ for triangle-free graphs with maximum degree at most $d$ and average degree at most $(d-2)(d^2-2d+2)/(d^2-3d+5)$.