AI 中文总结
本文针对不含K₄或K₂,₃子式的图，证明其HL指标R(G)的严格上界为√5-1，还给出最大度与平均度受限的无三角形图的R(G)上界，丰富了图的特征值相关理论。
AI 中文摘要
设λ₁≥λ₂≥…≥λₙ为n阶简单图G的特征值，G的HL指标定义为R(G)=max{|λₕ|,|λₗ|}，其中h=⌊(n+1)/2⌋，l=⌈(n+1)/2⌉。本文证明：若G不含K₄子式或不含K₂,₃子式，则R(G)≤√5-1，且无限族外平面图可达到等号；进一步，对于最大度不超过d、平均度不超过(d-2)(d²-2d+2)/(d²-3d+5)的无三角形图，R(G)≤√(d-2)。
英文摘要
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 $(d-2)(d^2-2d+2)/(d^2-3d+5)$.
Comments15 pages, 1 figure
