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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 $G$.

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 $d$., Comment: 30 pages. In this updated version, we have added a theorem concerning the rigidity of dense graphs and incorporated references to Vill\'anyi's recent resolution of the Lov\'asz-Yemini conjecture

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.

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

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 $A\setminus B$., Comment: 20 pages, 1 figure

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.

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.

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 $n\geq 2d$, \[ \left\lceil\frac{n}{2d}\right\rceil-2d+1\leq a_d(K_n) \leq \frac{2n}{3(d-1)}+\frac{1}{3}. \]

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 $d+1$.

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.

Let $X$ be a simplicial complex on vertex set $V$. We say that $X$ is $d$-representable if it is isomorphic to the nerve of a family of convex sets in $\mathbb{R}^d$. We define the $d$-boxicity of $X$ as the minimal $k$ such that $X$ can be written as the intersection of $k$ $d$-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of $X$ is a set $\tau\subset V$ such that $\tau\notin X$ but $\sigma\in X$ for any $\sigma\subsetneq \tau$. We prove that the $d$-boxicity of a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$ is at most $\left\lfloor\frac{1}{d+1}\binom{n}{d}\right\rfloor$. The bound is sharp: the $d$-boxicity of a simplicial complex whose set of missing faces form a Steiner $(d,d+1,n)$-system is exactly $\frac{1}{d+1}\binom{n}{d}$.

Let $G=(V,E)$ be a graph and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose simplices are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of the complexes $I_n(G)$ for various classes of graphs, focusing on the class of graphs with maximum degree bounded by $\Delta$. As an application, we obtain the following result: Let $G$ be a claw-free graph with maximum degree at most $\Delta$. Then, every collection of $\left\lfloor\left(\frac{\Delta}{2}+1\right)(n-1)\right\rfloor+1$ independent sets in $G$ has a rainbow independent set of size $n$.

Let $\mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$ with covering number at most $p$ is $\left(\binom{r+p}{r}-1\right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $\mathcal{F}\subset\mathcal{H}$ is $\frac{1}{2}\binom{2r}{r}$-collapsible.

Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $\sigma\notin X$ such that $\tau\in X$ for any $\tau\subsetneq \sigma$. 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.

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 $k\geq d$ 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 $d-1\leq j \leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Mart\'inez-Sandoval and Montejano for general position sets in matroids.