Alan Lew

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math.CO2026

An approximate version of Brouwer's Laplacian conjecture

Alan Lew

Let $G = (V, E)$ be an $n$ -vertex graph, $L (G) \in R^{n \times n}$ its Laplacian matrix, and let $I^»_{1} (L (G)) \geq I^»_{2} (L (G)) \geq \dots \geq I^»_{n} (L (G)) = 0$ denote its eigenvalues. Fo…

math.CO2025

An eigenvalue interlacing approach to Garland's method

Alan Lew

Let $X$ be a pure $d$ -dimensional simplicial complex. For $0 \leq k \leq d$ , let $X (k)$ be the set of $k$ -dimensional faces of $X$ , let $L ~_{k} (X)$ be the $k$ -dimensional weighted…

math.CO2025

Sums of Laplacian eigenvalues and sums of degrees

Alan Lew

Let $X$ be a simplicial complex. For $1 \leq i \leq dim (X)$ , let $X (i)$ be the set of $i$ -dimensional faces of $X$ , and let $f_{i} (X) = ∣ X (i) ∣$ . For $0 \leq i \leq dim (X) - 1$ , let $L_{i}^{+} (X)$ b…

math.CO2024

Partition density, star arboricity, and sums of Laplacian eigenvalues of graphs

Alan Lew

Let $G = (V, E)$ be a graph on $n$ vertices, and let $I^»_{1} (L (G)) \geq \dots \geq I^»_{n - 1} (L (G)) \geq I^»_{n} (L (G)) = 0$ be the eigenvalues of its Laplacian matrix $L (G)$ . Brouwer conjectured th…