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Spectral Gap Bounds for the Simplicial Laplacian and an Application To Spectral gap bounds for the simplicial Laplacian and an application to random complexes Samir Shukla,∗ D. Yogeshwaran† Abstract In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds generalize the result of Aharoni et al. (2005) [1] which is valid only for clique complexes. As an application, we investigate the thresholds for vanishing of cohomology of the neighborhood complex of the Erdös- Rényi random graph. We improve the upper bound derived in Kahle (2007) [15] by a logarithmic factor using our spectral gap bounds and we also improve the lower bound via finer probabilistic estimates than those in Kahle (2007) [15]. Keywords: spectral bounds, Laplacian, cohomology, neighborhood complex, Erdös-Rényi random graphs. AMS MSC 2010: 05C80. 05C50. 05E45. 55U10. 1 Introduction Let G be a graph with vertex set V (G) (often to be abbreviated as V ) and let L(G) denote the (unnormalized) Laplacian of G. Let 0 = λ1(G) ≤ λ2(G) ≤ . ≤ λ|V |(G) denote the eigenvalues of L(G) in ascending order. Here, the second smallest eigenvalue λ2(G) is called the spectral gap. The clique complex of a graph G is the simplicial complex whose simplices are all subsets σ ⊂ V which spans a complete subgraph of G. We shall denote the kth k arXiv:1810.10934v2 [math.CO] 9 Oct 2019 reduced cohomology of a simplicial complex X by H (X). In this article, all simplicial complexes are assumed to be connected and we always consider the reduced cohomology with real coefficients. For more detailed definitions, seee Section 2. In [1], Aharoni et al. proved that if the spectral gap of its 1-skeleton is large enough, then the cohomology of the corresponding clique complex vanishes. k|V | Theorem 1.1. [1, Theorem 1.2] Let X be the clique complex of a graph G. If λ2(G) > k+1 , then Hk(X) = 0. Aharonie et al. ([1]) used Theorem 1.1 to derive a lower bound for the homological connectivity of the independence complex of a graph G (a simplicial complex whose simplices ∗Department of Mathematics, Indian Institute of Technology Bombay, India. [email protected] †Statistics and Mathematics Unit, Indian Statistical Institute Bangaluru, India. [email protected] 1 are the independent sets of G) and which implies Hall type theorems for systems of disjoint representatives in hypergraphs. Theorem 1.1 can be viewed as a global counterpart for clique complexes of the spectral gap results of Garland ([9, Theorem 5.9]) and Ballman-Świ¸atkowski ([7, Theorem 2.5]) for vanishing of cohomology of a simplicial complex. In their simplest form, these results say that for a pure k-dimensional finite simplicial complex ∆, if the spectral gap of the link k−1 lk∆(τ) is sufficiently large for every (k − 2)-dimensional simplex τ, then H (∆) = 0. A very powerful application of the afore-mentioned result can be found in Kahle ([16]) where this was used to derive sharp vanishing thresholds for cohomology ofe random clique complexes. See [11] for more applications of this spectral gap result in random topology. Recently, Hino and Kanazawa ([10, Theorem 2.5]) generalized this result of Garland and Ballman-Świ¸atkowski, and upper bounded the (d−1)th Betti number of a pure d-dimensional simplicial complex ∆ by the sum (taken over all (d−2)-dimensional simplices τ) of the num- ber of ‘suitably small’ eigenvalues in the spectrum of the laplacian of the link lk∆(τ). They used this quantitative version of the spectral gap result to prove weak laws for (persistent) lifetime sums of randomly weighted clique and d-dimensional complexes. Motivated by such applications of spectral gap bounds to random complexes, we seek to generalize Theorem 1.1 to more general simplicial complexes. We achieve two different generalizations (see Corollaries 1.5 and 1.7) by comparing an arbitrary simplicial complex with a larger complex and the corresponding clique complex in two different ways. Our aim in exploring this generalization was to obtain vanishing threshold for cohomology in other random complex models. We use one of our generalizations to improve the vanishing threshold for cohomology of a random neighborhood complex (see Theorem 1.8) by a loga- rithmic factor. After the result, we also discuss why it is difficult to apply Garland’s method and hence a different spectral gap result is needed. By computing the probabilities involved more precisely than in [15], we also improve the lower bound by a polynomial factor. The paper is organised as follows. In Section 1.1, we introduce some notation, which we shall use in rest of the paper. In Section 1.2, we state our results, which relate the cohomology and spectral gap. In Section 1.3, we state the results about the neighborhood complexes of a random graph. We also discuss our improvements in relation to the results of [15]. We give the necessary preliminaries from graph theory and topology in Section 2 . Section 3 contains the proofs of the results stated in Sections 1.2 and 1.3. 1.1 Notation We shall use the following notations throughout this paper. Let X be a (simplicial) complex on n vertices. We denote by GX the 1-skeleton of X, i.e., GX is the graph whose vertices are the 0-dimensional simplices and the edges are the 1-dimensional simplices of X. We shall always assume GX to be connected. Let X(k) denote the set of all k-dimensional oriented simplices of X. X is said to be a clique complex if for all k ≥ 0, X(k) is the set of k (k + 1)-cliques in the graph GX . For k ≥−1, let C (X; R) denote the space of real valued k k+1 k-cochains of X. Let δk(X) : C (X; R) → C (X; R) denote the coboundary operator. ∗ ∗ For k ≥ 0, let δk(X) denote the adjoint of δk(X) and ∆k(X) := δk−1(X)δk−1(X)+ ∗ δk(X)δk(X) denote the (simplicial) Laplacian (see Section 2 for details). Let µk(X) denote the minimal eigenvalue of ∆k(X). Observe that λ2(GX ) = µ0(X). We again emphasize 2 that we consider reduced cohomology with real coefficients. We shall now define two ways to measure the difference between two complexes. The first compares a complex to its subcomplex whereas the second compares a complex X to the corresponding clique complex of GX . For k ≥ 1, define ′ ′ Sk(X, X ) := max |{τ ∈ X(k + 1) \ X (k + 1) | σ ⊂ τ}|, (1) σ∈X′(k) where X′ is a subcomplex of X. We shall now on use X to denote a complex and X′ to denote a subcomplex of X. For a simplex σ ∈ X, the link of σ is the complex defined as lkX (σ) := {η ∈ X | σ ∪ η ∈ X and σ ∩ η = ∅}. For k ≥ 1 and 1 ≤ j ≤ k + 1, define Dk(X, j) := max |{u | u∈ / lkX (σ) and ∃ exactly j vertices v1, . , vj ∈ σ such that σ∈X(k) u ∈ lkX (σ \ {vi}) ∀ 1 ≤ i ≤ j}|. (2) Remark 1.2. If the k-skeleton of X is a clique complex of GX and u ∈ lkX (σ \ {v}) ∩ lkX (σ \ {w}) for some {v, w}⊆ σ then u ∈ lkX (σ \ {v}) ∀ v ∈ σ, i.e., any (k + 1)-subset of σ ∪ {u} will be a k-simplex. Therefore, in this case Dk(X, j) = 0 for all 2 ≤ j ≤ k and Dk(X, k +1)= max |{w | w∈ / lkX (σ) and any (k + 1)-subset of σ ∪ {w} is a k-simplex}|. σ∈X(k) (3) Thus, if X is a clique complex then Dk(X, j) = 0 for all 2 ≤ j ≤ k + 1. 1.2 Spectral gap and cohomology We shall now present our two spectral gap results and corollaries that generalize Theorem 1.1. We first recall the following key theorem from [1] that was used to derive Theorem 1.1. Theorem 1.3. [1, Theorem 1.1] Let X be a clique complex. For k ≥ 1, kµk(X) ≥ (k + 1)µk−1(X) − |V (GX )|. We prove our first main spectral gap result by directly comparing the operators δk(X) ′ and δk(X ). Following the theorem, we state a simple corollary which generalizes Theorem 1.1.. Theorem 1.4. For every simplicial complex X and every subcomplex X′ of X, and for k ≥ 1, ′ ′ µk(X ) ≥ µk(X) − (k + 2)Sk(X, X ). (4) ′ Corollary 1.5. Let X be a clique complex and X have the same 1-skeleton as X i.e.,GX′ = kn k+2 ′ k ′ GX . If λ2(GX ) > k+1 + k+1 Sk(X, X ), then H (X ) = 0. e 3 ′ ′ If X = X, then Sk(X, X ) = 0 and so Corollary 1.5 implies Theorem 1.1. Now, we present our generalization of Theorem 1.3 using Dk(X, j)’s and another corollary that generalizes Theorem 1.1. Theorem 1.6. For any simplicial complex X and for k ≥ 1, k+1 kµk(X) ≥ (k + 1)µk−1(X) − n − (k(k +1)+ j)Dk(X, j). (5) Xj=2 Corollary 1.7. Let k-skeleton of X be same as that of the k-skeleton of the clique complex kn k of GX . If λ2(GX ) > k+1 + (k + 1)Dk(X, k + 1), then H (X) = 0. From Remark 1.2, we know that for a clique complex X, D (X, j) = 0 ∀ 2 ≤ j ≤ k + 1.
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