arXiv subjects

Alan Lew

Publications and source records attributed to Alan Lew.

At least 19 recordsLinked to original sources

arXiv2026

Rigidity of expanders and pseudorandom graphs

A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, the only continuous motions of the vertices preserving the distances between all pairs of adjacent vertices are those induced from the isometries of $\mathbb{R}^d$ (that is, translations and rotations of the whole graph). In this paper, we study rigidity properties of pseudorandom graphs. First, we consider $C$-expander graphs, a class of graphs recently studied in the context of Hamiltonicity of pseudorandom graphs. These are $n$-vertex graphs for which every vertex set $A$ of size smaller than $n/(2C)$ has a neighbourhood of size at least $C|A|$, and for every pair of disjoint sets $A,B$ of size at least $n/(2C)$ each, there is at least one edge between $A$ and $B$. We show that for every $C\ge 8$ and every integer $n\ge 9C$, every $n$-vertex $C$-expander is $\lfloor C/8\rfloor$-rigid. Next, we study $(n,r,\lambda)$-graphs, which are $n$-vertex $r$-regular graphs whose non-trivial adjacency eigenvalues are bounded in absolute value by $\lambda$. This is a well-known family of graphs, known to possess various pseudorandom properties. We prove that there exist absolute constants $c_1,c_2>0$ such that every $(n,r,\lambda)$-graph $G$ with $\lambda\le c_1r$ is $\lfloor c_2r\rfloor$-rigid. Our results are sharp up to the value of the universal constants involved, and they improve and extend previous work by the authors on the rigidity of random and pseudorandom graphs.

math.CO↗

arXiv2026

Combinatorial sufficient conditions for graph rigidity and applications to random graphs

A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, every edge-length preserving continuous motion of the vertices preserves the distances between all pairs of non-adjacent vertices as well. In this paper, we present several new results on the rigidity of random graphs. In particular, we show that there exists $c>0$ such that, for $p\ge 2 \log{n}/n$, the binomial random graph $G(n,p)$ is with high probability (whp) $\lfloor c n p\rfloor$-rigid. This is sharp up to the constant $c$, and complements recent results of Peled and Peleg (in the regime $p= o(n^{-1/2})$), and of Jord\'an, Liu, and Vill\'anyi (in the constant $p$ regime). Moreover, we show that for every fixed $d\ge 2$ and $r\ge 501d$, a random $r$-regular graph is whp $d$-rigid, and that for $100/n\le p\le 2\log{n}/n$, the binomial random graph $G(n,p)$ contains whp an $\lfloor np/251\rfloor$-rigid subgraph with at least $(1-e^{-np/2})n$ vertices. Both results are sharp up to the multiplicative constant. In addition, we present a new sufficient condition for rigidity in terms of the minimum codegree of the graph (the minimum number of common neighbours of a pair of vertices in the graph). A main tool in our arguments is a new combinatorial sufficient condition for rigidity, which provides a common generalization to Whiteley's vertex-splitting lemmas, and to the "rigid partitions" method, developed in works by Crapo, Lindemann, Lew, Nevo, Peled and Raz, and by the present authors.

math.CO↗

arXiv2026

An approximate version of Brouwer's Laplacian conjecture

Let $G=(V,E)$ be an $n$-vertex graph, $L(G)\in \mathbb{R}^{n\times n}$ its Laplacian matrix, and let $\lambda_1(L(G))\ge \lambda_2(L(G))\ge \cdots\ge \lambda_n(L(G))=0$ denote its eigenvalues. For $1\le k\le n$, let $\varepsilon_k(G)= \sum_{i=1}^k \lambda_i(L(G)) -|E|$. We show that for every $1\le k\le n$, \[ \varepsilon_k(G) \le \max_{U\subset V,\, |U|=k} |E_G(U)| + (4k-2)\sqrt{k}, \] where $E_G(U)$ is the set of edges of $G$ contained in $U$. As an immediate consequence, we obtain that $\varepsilon_k(G)\le \binom{k}{2}+(4k-2)\sqrt{k}$. This improves upon previously known bounds for large values of $k$, and may be seen as an approximate version of a conjecture of Brouwer, stating that $\varepsilon_k(G)\le \binom{k+1}{2}$ for every graph $G$. Moreover, for every $r\ge 2$, if $G$ is a $K_{r+1}$-free graph, we obtain that $\varepsilon_k(G)\le (1-1/r)k^2/2 + (4k-2)\sqrt{k}$, which is tight up to the sub-quadratic term. Our arguments rely on the study of the largest eigenvalue of a matrix obtained by performing a certain diagonal perturbation on the $k$-th additive compound matrix of $L(G)$. Using similar methods, we show that the largest Laplacian eigenvalue of the $k$-th token graph of a graph $G=(V,E)$ is bounded from above by $|E|+4k-2$, obtaining a weak version of a conjecture of Apte, Parekh, and Sud, which predicts that an upper bound of $|E|+k$ should hold. All our results also hold, with essentially the same proofs, when the Laplacian matrix is replaced by the signless Laplacian of the graph.

math.CO↗

arXiv2025

An eigenvalue interlacing approach to Garland's method

Let $X$ be a pure $d$-dimensional simplicial complex. For $0\le k\le d$, let $X(k)$ be the set of $k$-dimensional faces of $X$, let $\tilde{L}_k(X)$ be the $k$-dimensional weighted total Laplacian operator on $X$, and let $\tilde{H}_k(X;\mathbb{R})$ be its $k$-dimensional reduced homology group with real coefficients. For $\sigma\in X$, let $\text{lk}(X,\sigma)$ be the link of $\sigma$ in $X$. For a matrix $M$, we denote by $\text{Spec}(M)$ the multi-set containing all the eigenvalues of $M$. We show that, for every $0\le \ell<k \le d$, \[ \text{dim}(\tilde{H}_k(X;\mathbb{R}))\le \sum_{\eta\in X(\ell)}\left| \left\{ \lambda\in \text{Spec}(\tilde{L}_{k-\ell-1}(\text{lk}(X,\eta))) :\, \lambda\le \frac{(\ell+1)(d-k)}{k+1}\right\}\right|. \] This extends the classical vanishing theorem of Garland, corresponding to the special case when the right hand side of the inequality is equal to zero, and a more recent result by Hino and Kanazawa, corresponding to the case $\ell=k-1$. A main new ingredient in our proof is an abstract version of Garland's local to global principle, which follows as a simple consequence of the eigenvalue interlacing theorem, and may be of independent interest.

math.CO↗

arXiv2025

Sums of Laplacian eigenvalues and sums of degrees

Let $X$ be a simplicial complex. For $1\le i\le\dim(X)$, let $X(i)$ be the set of $i$-dimensional faces of $X$, and let $f_i(X)=|X(i)|$. For $0\le i\le \dim(X)-1$, let $L_i^+(X)$ be the $i$-th upper Laplacian operator of $X$. For $\sigma\in X$ and $1\le r\le \dim(X)$, we denote by $\text{deg}_X^{(r)}(\sigma)$ the number of $r$-dimensional faces of $X$ containing $\sigma$. For a symmetric matrix $M\in \mathbb{R}^{n\times n}$ and $1\le i\le n$, let $\lambda_i(M)$ be the $i$-th largest eigenvalue of $M$. We prove that for every complex $X$, $1\le r\le\dim(X)$, and $1\le k\le f_{r-1}(X)/(r+1)$, \[ \sum_{i=1}^k \lambda_i(L_{r-1}^+(X)) \le \max \left\{ \sum_{\sigma\in A} \text{deg}_X^{(r)}(\sigma) :\, A\subset X(r-1),\, |A|=(r+1)k \right\}. \] This bound is sharp, and it extends a classical result of Anderson and Morley, corresponding to the special case $k=1,\, r=1$. As a consequence, we show that for all $1\le r\le \dim(X)$ and $1\le k\le f_{r-1}(X)$, \[ \sum_{i=1}^{k} \lambda_i(L_{r-1}^+(X)) \le f_r(X) + \binom{(r+1)k}{2}. \] In the case $r=1$, we obtain the following improved bound: for every $k\ge 1$ and every graph $G=(V,E)$ with $|V|\ge k$, \[ \sum_{i=1}^k \lambda_i(L(G)) \leq |E|+k^2, \] where $L(G)=L_0^{+}(G)$ is the Laplacian matrix of $G$. This improves upon previously known bounds for all $k\ge 3$, and may be seen as a further step towards Brouwer's conjecture, which states that $\sum_{i=1}^k \lambda_i(L(G)) \leq |E|+\binom{k+1}{2}.$ As an additional application, we show that if $X$ is an $(r+1)$-partite $r$-dimensional simplicial complex on vertex set $V$, and $1\le k\le f_{r-1}(X)$, then \[ \sum_{i=1}^{k} \lambda_i(L_{r-1}^+(X)) \le \sum_{i=1}^k \left|\{v\in V:\, \text{deg}^{(r)}_X(v)\ge i\}\right|. \] This resolves a special case of a conjecture of Duval and Reiner, which states that the above inequality holds for all simplicial complexes.

math.CO↗

arXiv2025

Stiffness matrices of graph blow-ups and the $d$-dimensional algebraic connectivity of complete bipartite graphs

The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$ is a quantitative measure of its $d$-dimensional rigidity, defined in terms of the eigenvalues of stiffness matrices associated with different embeddings of the graph into $\mathbb{R}^d$. For a function $a:V\to \mathbb{N}$, we denote by $G^{(a)}$ the $a$-blow-up of $G$, that is, the graph obtained from $G$ by replacing every vertex $v\in V$ with an independent set of size $a(v)$. We determine a relation between the stiffness matrix eigenvalues of $G^{(a)}$ and the eigenvalues of certain weighted stiffness matrices associated with the original graph $G$. This resolves, as a special case, a conjecture of Lew, Nevo, Peled and Raz on the stiffness eigenvalues of balanced blow-ups of the complete graph. As an application, we obtain a lower bound on the $d$-dimensional algebraic connectivity of complete bipartite graphs. More precisely, we prove the following: Let $K_{n,m}$ be the complete bipartite graph with sides of size $n$ and $m$ respectively. Then, for every $d\ge 1$ there exists $c_d>0$ such that, for all $n,m\ge d+1$ with $n+m\ge \binom{d+2}{2}$, $a_d(K_{n,m})\ge c_d\cdot \min\{n,m\}$. This bound is tight up to the multiplicative constant. In the special case $d=2$, $n=m=3$, we obtain the improved bound $a_2(K_{3,3})\ge 2(1-\lambda)$, where $\lambda\approx 0.6903845$ is the unique positive real root of the polynomial $176 x^4-200 x^3+47 x^2+18 x-9$, which we conjecture to be tight.

math.CO↗

arXiv2025

On the $k$-volume rigidity of a simplicial complex in $\mathbb{R}^d$

We define a generic rigidity matroid for $k$-volumes of a simplicial complex in $\mathbb{R}^d$, and prove that for $2\leq k \leq d-1$ it has the same rank as the classical generic $d$-rigidity matroid on the same vertex set (namely, the case $k=1$). This is in contrast with the $k=d$ case, previously studied by Lubetzky and Peled, which presents a different behavior. We conjecture a characterization for the bases of this matroid in terms of $d$-rigidity of the $1$-skeleton of the complex and a combinatorial Hall condition on incidences of edges in $k$-faces.

math.CO↗

arXiv2024

Minimum degree conditions for graph rigidity

We study minimum degree conditions that guarantee that an $n$-vertex graph is rigid in $\mathbb{R}^d$. For small values of $d$, we obtain a tight bound: for $d = O(\sqrt{n})$, every $n$-vertex graph with minimum degree at least $(n+d)/2 - 1$ is rigid in $\mathbb{R}^d$. For larger values of $d$, we achieve an approximate result: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $(n+2d)/2 - 1$ is rigid in $\mathbb{R}^d$. This bound is tight up to a factor of two in the coefficient of $d$. As a byproduct of our proof, we also obtain the following result, which may be of independent interest: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $d$ has pseudoachromatic number at least $d+1$; namely, the vertex set of such a graph can be partitioned into $d+1$ subsets such that there is at least one edge between each pair of subsets. This is tight.

math.CO↗

arXiv2024

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

Let $G=(V,E)$ be a graph on $n$ vertices, and let $\lambda_1(L(G))\ge \cdots\ge \lambda_{n-1}(L(G))\ge \lambda_n(L(G))=0$ be the eigenvalues of its Laplacian matrix $L(G)$. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k \lambda_i(L(G)) \le |E|+\binom{k+1}{2}$. Here, we prove the following weak version of Brouwer's conjecture: For every $1\leq k \leq n$, \[ \sum_{i=1}^k \lambda_i(L(G)) \leq |E|+k^2+15k\log{k}+65k. \] For a graph $G=(V,E)$, we define its partition density $\tilde{\rho}(G)$ as the maximum, over all subgraphs $H$ of $G$, of the ratio between the number of edges of $H$ and the number of vertices in the largest connected component of $H$. Our argument relies on the study of the structure of the graphs $G$ satisfying $\tilde{\rho}(G)< k$. In particular, using a result of Alon, McDiarmid and Reed, we show that every such graph can be decomposed into at most $k+ 15\log{k}+65$ edge-disjoint star forests (that is, forests whose connected components are all isomorphic to stars). In addition, we show that for every graph $G=(V,E)$ and every $1\le k\le |V|$, \[ \sum_{i=1}^k \lambda_i(L(G)) \leq |E|+k\cdot \nu(G) + \left\lfloor\frac{k}{2}\right\rfloor, \] where $\nu(G)$ is the maximum size of a matching in $G$.

math.CO↗

arXiv2023

Rigid partitions: from high connectivity to random graphs

A graph is called $d$-rigid if there exists a generic embedding of its vertex set into $\mathbb{R}^d$ such that every continuous motion of the vertices that preserves the lengths of all edges actually preserves the distances between all pairs of vertices. The rigidity of a graph is the maximal $d$ such that the graph is $d$-rigid. We present new sufficient conditions for the $d$-rigidity of a graph in terms of the existence of ``rigid partitions'' -- partitions of the graph that satisfy certain connectivity properties. This extends previous results by Crapo, Lindemann, and Lew, Nevo, Peled and Raz. As an application, we present new results on the rigidity of highly-connected graphs, random graphs, random bipartite graphs, pseudorandom graphs, and dense graphs. In particular, we prove that random $C d\log d$-regular graphs are typically $d$-rigid, demonstrate the existence of a giant $d$-rigid component in sparse random binomial graphs, and show that the rigidity of relatively sparse random binomial bipartite graphs is roughly the same as that of the complete bipartite graph, which we consider an interesting phenomenon. Furthermore, we show that a graph admitting $\binom{d+1}{2}$ disjoint connected dominating sets is $d$-rigid. This implies a weak version of the Lov\'asz--Yemini conjecture on the rigidity of highly-connected graphs. We also present an alternative short proof for a recent result by Lew, Nevo, Peled, and Raz, which asserts that the hitting time for $d$-rigidity in the random graph process typically coincides with the hitting time for minimum degree $d$.

math.CO↗

arXiv2023

Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs

Let $\text{Fl}_{n,q}$ be the simplicial complex whose vertices are the non-trivial subspaces of $\mathbb{F}_q^n$ and whose simplices correspond to families of subspaces forming a flag. Let $\Delta^{+}_k(\text{Fl}_{n,q})$ be the $k$-dimensional weighted upper Laplacian on $ \text{Fl}_{n,q}$. The spectrum of $\Delta^{+}_k(\text{Fl}_{n,q})$ was first studied by Garland, who obtained a lower bound on its non-zero eigenvalues. Here, we focus on the $k=0$ case. We determine the asymptotic behavior of the eigenvalues of $\Delta_{0}^{+}(\text{Fl}_{n,q})$ as $q$ tends to infinity. In particular, we show that for large enough $q$, $\Delta_{0}^{+}(\text{Fl}_{n,q})$ has exactly $\left\lfloor n^2/4\right\rfloor+2$ distinct eigenvalues, and that every eigenvalue $\lambda\neq 0,n-1$ of $\Delta_{0}^{+}(\text{Fl}_{n,q})$ tends to $n-2$ as $q$ goes to infinity. This solves the $0$-dimensional case of a conjecture of Papikian.

math.CO↗

arXiv2023

Laplacian eigenvalues of independence complexes via additive compound matrices

The independence complex of a graph $G=(V,E)$ is the simplicial complex $I(G)$ on vertex set $V$ whose simplices are the independent sets in $G$. We present new lower bounds on the eigenvalues of the $k$-dimensional Laplacian $L_k(I(G))$ in terms of the eigenvalues of the graph Laplacian $L(G)$. As a consequence, we show that for all $k\geq 0$, the dimension of the $k$-th reduced homology group (with real coefficients) of $I(G)$ is at most \[ \left| \left\{ 1\leq i_1<\cdots<i_{k+1}\leq |V| : \, \lambda_{i_1}+\lambda_{i_2}+\cdots+\lambda_{i_{k+1}} \geq |V|\right\}\right|,\] where $\lambda_1\geq\lambda_2\geq \cdots\geq \lambda_{|V|}=0$ are the eigenvalues of $L(G)$. In particular, if $k$ is the minimal number such that the sum of the $k$ largest eigenvalues of $L(G)$ is at least $|V|$, then $\tilde{H}_i(I(G);\mathbb{R})=0$ for all $i\leq k-2$. This extends previous results by Aharoni, Berger and Meshulam. Our proof relies on a relation between the $k$-dimensional Laplacian $L_k(I(G))$ and the $(k+1)$-th additive compound matrix of $L_0(I(G))$, which is an $\binom{n}{k+1}\times\binom{n}{k+1}$ matrix whose eigenvalues are all the possible sums of $k+1$ eigenvalues of the $0$-dimensional Laplacian. Our results apply also in the more general setting of vertex-weighted Laplacian matrices.

math.CO↗

arXiv2023

Extensions of the Colorful Helly Theorem for $d$-collapsible and $d$-Leray complexes

We present extensions of the Colorful Helly Theorem for $d$-collapsible and $d$-Leray complexes, providing a common generalization to the matroidal versions of the theorem due to Kalai and Meshulam, the ``very colorful" Helly theorem introduced by Arocha, B\'ar\'any, Bracho, Fabila and Montejano, and the ``semi-intersecting" colorful Helly theorem proved by Montejano and Karasev. As an application, we obtain the following extension of Tverberg's Theorem: Let $A$ be a finite set of points in $\mathbb{R}^d$ with $|A|>(r-1)(d+1)$. Then, there exist a partition $A_1,\ldots,A_r$ of $A$ and a subset $B\subset A$ of size $(r-1)(d+1)$, such that $\cap_{i=1}^r \text{conv}( (B\cup\{p\})\cap A_i)\neq\emptyset$ for all $p\in A\setminus B$. That is, we obtain a partition of $A$ into $r$ parts that remains a Tverberg partition even after removing all but one arbitrary point from $A\setminus B$.

math.CO↗

arXiv2023

Garland's method for token graphs

The $k$-th token graph of a graph $G=(V,E)$ is the graph $F_k(G)$ whose vertices are the $k$-subsets of $V$ and whose edges are all pairs of $k$-subsets $A,B$ such that the symmetric difference of $A$ and $B$ forms an edge in $G$. Let $L(G)$ be the Laplacian matrix of $G$, and $L_k(G)$ be the Laplacian matrix of $F_k(G)$. It was shown by Dalf\'o et al. that for any graph $G$ on $n$ vertices and any $0\leq \ell \leq k \leq \left\lfloor n/2\right\rfloor$, the spectrum of $L_{\ell}(G)$ is contained in that of $L_k(G)$. Here, we continue to study the relation between the spectrum of $L_k(G)$ and that of $L_{k-1}(G)$. In particular, we show that, for $1\leq k\leq \left\lfloor n/2\right\rfloor$, any eigenvalue $\lambda$ of $L_k(G)$ that is not contained in the spectrum of $L_{k-1}(G)$ satisfies \[ k(\lambda_2(L(G))-k+1)\leq \lambda \leq k\lambda_n(L(G)), \] where $\lambda_2(L(G))$ is the second smallest eigenvalue of $L(G)$ (a.k.a. the algebraic connectivity of $G$), and $\lambda_n(L(G))$ is its largest eigenvalue. Our proof relies on an adaptation of Garland's method, originally developed for the study of high-dimensional Laplacians of simplicial complexes.

math.CO↗

arXiv2023

Rigidity expander graphs

Jord\'an and Tanigawa recently introduced the $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G$. This is a quantitative measure of the $d$-dimensional rigidity of $G$ which generalizes the well-studied notion of spectral expansion of graphs. We present a new lower bound for $a_d(G)$ defined in terms of the spectral expansion of certain subgraphs of $G$ associated with a partition of its vertices into $d$ parts. In particular, we obtain a new sufficient condition for the rigidity of a graph $G$. As a first application, we prove the existence of an infinite family of $k$-regular $d$-rigidity-expander graphs for every $d\ge 2$ and $k\ge 2d+1$. Conjecturally, no such family of $2d$-regular graphs exists. Second, we show that $a_d(K_n)\geq \frac{1}{2}\left\lfloor\frac{n}{d}\right\rfloor$, which we conjecture to be essentially tight. In addition, we study the extremal values $a_d(G)$ attained if $G$ is a minimally $d$-rigid graph.

math.CO↗

arXiv2022

On the $d$-dimensional algebraic connectivity of graphs

The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$, introduced by Jord\'an and Tanigawa, is a quantitative measure of the $d$-dimensional rigidity of $G$ that is defined in terms of the eigenvalues of stiffness matrices (which are analogues of the graph Laplacian) associated to mappings of the vertex set $V$ into $\mathbb{R}^d$. Here, we analyze the $d$-dimensional algebraic connectivity of complete graphs. In particular, we show that, for $d\geq 3$, $a_d(K_{d+1})=1$, and for $n\geq 2d$, \[ \left\lceil\frac{n}{2d}\right\rceil-2d+1\leq a_d(K_n) \leq \frac{2n}{3(d-1)}+\frac{1}{3}. \]

math.CO↗

arXiv2022

Sharp threshold for rigidity of random graphs

We consider the Erd\H{o}s-R\'enyi evolution of random graphs, where a new uniformly distributed edge is added to the graph in every step. For every fixed $d\ge 1$, we show that with high probability, the graph becomes rigid in $\mathbb R^d$ at the very moment its minimum degree becomes $d$, and it becomes globally rigid in $\mathbb R^d$ at the very moment its minimum degree becomes $d+1$.

math.CO↗

arXiv2021

Leray numbers of tolerance complexes

Let $K$ be a simplicial complex on vertex set $V$. $K$ is called $d$-Leray if the homology groups of any induced subcomplex of $K$ are trivial in dimensions $d$ and higher. $K$ is called $d$-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most $d$ that is contained in a unique maximal face. We define the $t$-tolerance complex of $K$, $\mathcal{T}_t(K)$, as the simplicial complex on vertex set $V$ whose simplices are formed as the union of a simplex in $K$ and a set of size at most $t$. We prove that for any $d$ and $t$ there exists a positive integer $h(t,d)$ such that, for every $d$-collapsible complex $K$, the $t$-tolerance complex $\mathcal{T}_t(K)$ is $h(t,d)$-Leray. The definition of the complex $\mathcal{T}_t(K)$ is motivated by results of Montejano and Oliveros on "tolerant" versions of Helly's theorem. As an application, we present some new tolerant versions of the colorful Helly theorem.

math.CO↗