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Register a new userSpectral gaps of simplicial complexes without large missing faces
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Authors
Alan Lew
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Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{j}$ denote the $j$-Laplacian acting on real $j$-cochains of $X$ and let $mu_{j}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $mu_{k}(X)$ for $kgeq d$ and $mu_{d-1}(X)$. In particular, we establish the following vanishing result: If $mu_{d-1}(X)>(1-binom{k+1}{d}^{-1})n$, then $tilde{H}^{j}(X;mathbb{R})=0$ for all $d-1leq j leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martinez-Sandoval and Montejano for general position sets in matroids.
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Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $sigma otin X$ such that $tauin X$ for any $tausubsetneq sigma$. For a $k$-dimensional simplex $sigma$ in $X$, its degree in $X$ is the number of $(k+1)$-dimensional simplices in $X$ containing it. Let $delta_k$ denote the minimal degree of a $k$-dimensional simplex in $X$. Let $L_k$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $mu_k(X)$ denote its minimal eigenvalue. We prove the following lower bound on the spectral gaps $mu_k(X)$, for complexes $X$ without missing faces of dimension larger than $d$: [ mu_k(X)geq (d+1)(delta_k+k+1)-d n. ] As a consequence we obtain a new proof of a vanishing result for the homology of simplicial complexes without large missing faces. We present a family of examples achieving equality at all dimensions, showing that the bound is tight. For $d=1$ we characterize the equality case.
We show that the size of the largest simple d-cycle in a simplicial d-complex $K$ is at least a square root of $K$s density. This generalizes a well-known classical result of ErdH{o}s and Gallai cite{EG59} for graphs. We use methods from matroid theory applied to combinatorial simplicial complexes.
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