单纯复形的非零上拉普拉斯特征值重数上界

中文摘要 AI 辅助

设 $K$ 为具有至少两个面的有限纯脊连通 $d$ 维单纯复形，并设 $m_K(\lambda)$ 表示其上拉普拉斯算子 $L_{d-1}^{\mathrm{up}}(K)$ 的正特征值 $\lambda$ 的重数。我们根据基本组合参数 $f_d(K)$ 和 $f_{d-1}(K)$（面和脊的数量）、$p_d(K)$（悬挂面的数量）以及 $q_d(K)$（准悬挂脊的数量）获得了 $m_K(\lambda)$ 的上界。量 $\theta_d(K)=d f_d(K)-f_{d-1}(K)+1$ 作为自然约化面-脊关联图的圈数出现。我们证明 \\[ m_K(\lambda)\le 2\theta_d(K)+p_d(K), \qquad m_K(\lambda)\le 2\theta_d(K)+q_d(K)\quad(\lambda\ne d). \\] 我们还刻画了在这些界中达到等号的复形。最后，对于每个有限纯 $d$ 维单纯复形，我们证明了尖锐不等式 $m_K(d)\ge p_d(K)-q_d(K)$，这是 Faria 不等式的高维类比。对于 $d=1$，我们的结果归结为图拉普拉斯算子的相应重数界。

英文摘要

Let $K$ be a finite pure ridge-connected $d$-dimensional simplicial complex with at least two facets, and let $m_K(λ)$ denote the multiplicity of a positive eigenvalue $λ$ of its upper Laplacian $L_{d-1}^{\mathrm{up}}(K)$. We obtain upper bounds for $m_K(λ)$ in terms of the basic combinatorial parameters $f_d(K)$ and $f_{d-1}(K)$, the numbers of facets and ridges, $p_d(K)$, the number of pendant facets, and $q_d(K)$, the number of quasi-pendant ridges. The quantity $θ_d(K)=d f_d(K)-f_{d-1}(K)+1$ arises as the cyclomatic number of a natural reduced facet--ridge incidence graph. We prove \[ m_K(λ)\le 2θ_d(K)+p_d(K), \qquad m_K(λ)\le 2θ_d(K)+q_d(K)\quad(λ\ne d). \] We also characterize the complexes for which equality is attained in these bounds. Finally, for every finite pure $d$-dimensional simplicial complex, we prove the sharp inequality $m_K(d)\ge p_d(K)-q_d(K)$, which is a higher-dimensional analogue of Faria's inequality. For $d=1$, our results reduce to the corresponding multiplicity bounds for graph Laplacians.

发表机构

• Indian Institute of Technology Guwahati（印度理工学院古瓦哈提分校）

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