Combinatorial Laplacians Research Articles

Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs

We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph’s genus g . These bounds can be further improved if g=0 , i.e., if the metric graph is planar. Our results are based on a spectral correspondence between the Kirchhoff Laplacian and a particular combinatorial weighted Laplacian. In order to take advantage of this correspondence, we also prove new estimates for the eigenvalues of the weighted combinatorial Laplacians that were previously known only in the unweighted case.

HOMOTOPY CONTINUATION FOR THE SPECTRA OF PERSISTENT LAPLACIANS.

The p-persistent q-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the q-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.

The theta number of simplicial complexes

We introduce a generalization of the celebrated Lovasz theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such as the relationship to Hoffman’s ratio bound and to the chromatic number and study how they extend to higher dimensions. Like in the case of graphs, the higher dimensional theta number can be extended to a hierarchy of semidefinite programming upper bounds reaching the independence number. We analyze the value of the theta number and of the hierarchy for dense random simplicial complexes.

Symmetric Laplacians, quantum density matrices and their Von-Neumann entropy

We show that the (normalized) symmetric Laplacian of a simple graph can be obtained from the partial trace over a pure bipartite quantum state that resides in a bipartite Hilbert space (one part corresponding to the vertices, the other corresponding to the edges). This suggests an interpretation of the symmetric Laplacian's Von Neumann entropy as a measure of bipartite entanglement present between the two parts of the state. We then study extreme values for a connected graph's generalized Rényi-p entropy. Specifically, we show that(1)the complete graph achieves maximum entropy,(2)the 2-regular graph:(a)achieves minimum Rényi-2 entropy among all k-regular graphs,(b)is within log⁡4/3 of the minimum Rényi-2 entropy and log⁡42/3 of the minimum Von Neumann entropy among all connected graphs,(c)achieves a Von Neumann entropy less than the star graph. Point (2) contrasts sharply with similar work applied to (normalized) combinatorial Laplacians, where it has been shown that the star graph almost always achieves minimum Von Neumann entropy. In this work we find that the star graph achieves maximum entropy in the limit as the number of vertices grows without bound.

Hamiltonian Cycle Enumeration via Fermion-Zeon Convolution

Operators are induced on fermion and zeon algebras by the action of adjacency matrices and combinatorial Laplacians on the vector spaces spanned by the graph's vertices. Properties of the algebras automatically give information about the graph's spanning trees and vertex coverings by cycles \& matchings. Combining the properties of operators induced on fermions and zeons gives a fermion-zeon convolution that recovers the number of Hamiltonian cycles in an arbitrary graph. The mathematics underlying the graph-theoretic interpretation of these operators is provided by Kirchhoff's theorem and by the seminal works of Goulden and Jackson and Liu, who established formulas for enumeration of Hamiltonian cycles and paths using determinants and permanents of adjacency matrices.

A formula for simplicial tree-numbers of matroid complexes

We give a formula for the simplicial tree-numbers of the independent set complex of a finite matroid M as a product of eigenvalues of the total combinatorial Laplacians on this complex. Two matroid invariants emerge naturally in describing the multiplicities of these eigenvalues in the formula: one is the unsigned reduced Euler characteristic of the independent set complex and the other is the β-invariant of a matroid. We will demonstrate various applications of this formula including a “matroid theoretic” derivation of Kalai’s simplicial tree-numbers of a standard simplex.

Weighted Tree-Numbers of Matroid Complexes

We give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s $\beta$-invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes. Nous présentons une nouvelle formule pour les nombres d’arbres pondérés de grande dimension des matroïdes complexes. Cette formule est dérivée du résultat que le spectre des Laplaciens combinatoires pondérés des matrides complexes sont des polynômes à plusieurs variables. Dans la formule, le $\beta$;-invariant de Crapo apparaît comme étant le facteur clé reliant les Laplaciens combinatoires pondérés et les nombres d’arbres pondérés des matroïdes complexes.

Discrete Laplacians on general polygonal meshes

While the theory and applications of discrete Laplacians on triangulated surfaces are well developed, far less is known about the general polygonal case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non-planar and non-convex polygons. Our construction is guided by closely mimicking structural properties of the smooth Laplace--Beltrami operator. Among other features, our construction leads to an extension of the widely employed cotan formula from triangles to polygons. Besides carefully laying out theoretical aspects, we demonstrate the versatility of our approach for a variety of geometry processing applications, embarking on situations that would have been more difficult to achieve based on geometric Laplacians for simplicial meshes or purely combinatorial Laplacians for general meshes.

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.

Combinatorial laplacians and positivity under partial transpose

The density matrices of graphs are combinatorial laplacians normalised to have trace one (Braunsteinet al. 2006b). If the vertices of a graph are arranged as an array, its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for the separability of density matrices of graphs, is equivalent to the PPT-criterion. As such, it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Nonetheless, we prove the sufficiency when one of the array dimensions has length two (see Wu (2006) for an alternative proof). Finally, we derive a rational upper bound on the concurrence of density matrices of graphs and show that this bound is exact for graphs on four vertices.

The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States

We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties are linked to the entanglement of the state. We calculate the concurrence of all graph on four vertices representing entangled states. It turns out that for some of these graphs the value of the concurrence is exactly fractional.

Combinatorial Laplacians of matroid complexes

We combinatorially interpret the spectra of discrete Laplace operators from the boundary maps in the simplicial complex of independent sets of a matroid. The interpretation follows from a surprising orthogonal decomposition of the simplicial chain groups. This decomposition is in general finer than the spectral decomposition. As a consequence, the spectra are integral. One corollary to our combinatorial interpretation may be paraphrased as stating that one can “hear" the characteristic polynomial of a matroid.

Von Neumann spectra near the spectral gap

In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L 2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L 2 theta and L 2 zera functions defined by metric dependent combinatorial Laplacians acting on L 2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.

On Isospectral Graphs.

We study combinatorial Laplacians of graphs, and provide examples of isospectral graphs, including finite and infinite ones, as well as isospectral graphs with boundaries. Two general methods for the construction of isospectral graphs with boundaries are also presented.

Survey of Spectra of Laplacians on Finite Symmetric Spaces

We surveywhat is known about spectra of combinatorial Laplacians (or adjacency operators) of graphs on the simplest finite symmetric spaces. This work is joint with J. Angel, N. Celniker, A. Medrano, P. Myers, S. Poulos, H. Stark, C. Trimble, and E.Velasquez. For each finite field Fq with q odd, we consider graphs associated to finite Euclidean and non-Euclidean symmetric spaces over Fq. We are mainly interested in three questions regarding the eigenvalues and eigenfunctions of the combinatorial Laplacian as q goes to infinity: How large is the second largest eigenvalue, in absolute value, compared with the graph's degree? (The largesteigenvalue is the degree.) What can one say about the distribution of eigenvalues? What can one say about the “level curves” of the eigenfunctions?

Determinants of Laplacians on graphs

IN RECENT years, it has been observed that the determinant of a Laplacian on a manifold can often be expressed in terms of the closed orbits of a flow on that (or a related) manifold. Examples include the work of many authors (see, for example [2], [3], [6], [14], [16], [19], [20]) which uses the Selberg trace formula to express the determinant of a Laplacian acting on sections of a vector bundle over a Riemann surface in terms of the closed geodesics on the surface. Other examples can be found in the work of D. Fried relating Reidemeister Torsion (a combinatorial invariant of finite C-W complexes which can be expressed in terms of determinants of combinatorial Laplacians) to the closed orbits of flows of various types ([S], [6], [7]). In this paper we show that this relationship is, in fact, fundamental, and can be seen on the level of graphs. Let G be a finite graph (all terms will be defined precisely in Section 1) with a weight attached to each vertex and edge. These weights induce an inner product on the spaces V* (the complex functions on the set of vertices of G) and E* (the complex functions on the set of edges). If 6 is the usual coboundary operator from V* to E* we define a (combinatorial) Laplacian A by A = 6*6

The appearance of critical dimensions in regulated string theories (II)

We supply numerical evidence for the existence of critical dimensions d c 1 and d c 2 gaussian model of a discretized string, between which the mean number of vertices in the world sheet diverges, and hence a continuum limit may exist. We also discuss the possibility of non-trivial continuum limits. In particular, by introducing an additional parameter into the model, we argue that non-trivial continuum limits can possibly be obtained only between two critical dimensions d′ 0 and d 0. Furthermore, we give proofs of some lower bounds on determinants of combinatorial laplacians entering the models, which were announced in a previous paper.