Computing the Permanent of the Laplacian Matrices of Nonbipartite Graphs

We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the Erdös–Rényi random graph G(n, pn) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, pn ∈ (0, 1) and npn → ∞, we prove that the empirical distribution of the eigenvalues of the Laplacian matrix converges to a deterministic distribution, which is the free convolution of the semi-circle law and N(0, 1). However, for its normalized version, we prove that the empirical distribution converges to the semi-circle law.

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Multi-View Spectral Clustering with Optimal Neighborhood Laplacian Matrix

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v34i04.6180 ◽

2020 ◽

Vol 34 (04) ◽

pp. 6965-6972

Author(s):

Sihang Zhou ◽

Xinwang Liu ◽

Jiyuan Liu ◽

Xifeng Guo ◽

Yawei Zhao ◽

...

Keyword(s):

Spectral Clustering ◽

Optimization Problem ◽

State Of The Art ◽

Laplacian Matrix ◽

High Order ◽

Insufficient Information ◽

Complementary Information ◽

First Order ◽

Laplacian Matrices ◽

Group Data

Multi-view spectral clustering aims to group data into different categories by optimally exploring complementary information from multiple Laplacian matrices. However, existing methods usually linearly combine a group of pre-specified first-order Laplacian matrices to construct an optimal Laplacian matrix, which may result in limited representation capability and insufficient information exploitation. In this paper, we propose a novel optimal neighborhood multi-view spectral clustering (ONMSC) algorithm to address these issues. Specifically, the proposed algorithm generates an optimal Laplacian matrix by searching the neighborhood of both the linear combination of the first-order and high-order base Laplacian matrices simultaneously. This design enhances the representative capacity of the optimal Laplacian and better utilizes the hidden high-order connection information, leading to improved clustering performance. An efficient algorithm with proved convergence is designed to solve the resultant optimization problem. Extensive experimental results on 9 datasets demonstrate the superiority of our algorithm against state-of-the-art methods, which verifies the effectiveness and advantages of the proposed ONMSC.

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Linear Software Models: An Occam’s Razor Set of Algebraic Connectors Integrates Modules into a Whole Software System

International Journal of Software Engineering and Knowledge Engineering ◽

10.1142/s0218194020400185 ◽

2020 ◽

Vol 30 (10) ◽

pp. 1375-1413

Author(s):

Iaakov Exman ◽

Harel Wallach

Keyword(s):

Spectral Method ◽

Algebraic Theory ◽

Laplacian Matrix ◽

Software Systems ◽

Software System ◽

Laplacian Matrices ◽

Software Models ◽

Occam’S Razor ◽

Software Modularity ◽

Occam's Razor

Well-designed software systems, with providers only modules, have been rigorously obtained by algebraic procedures from the software Laplacian Matrices or their respective Modularity Matrices. However, a complete view of the whole software system should display, besides provider relationships, also consumer relationships. Consumers may have two different roles in a system: either internal or external to modules. Composite modules, including both providers and internal consumers, are obtained from the joint providers and consumers Laplacian matrix, by the same spectral method which obtained providers only modules. The composite modules are integrated into a whole Software System by algebraic connectors. These algebraic connectors are a minimal Occam’s razor set of consumers external to composite modules, revealed through iterative splitting of the Laplacian matrix by Fiedler eigenvectors. The composite modules, of the respective standard Modularity Matrix for the whole software system, also obey linear independence of their constituent vectors, and display block-diagonality. The spectral method leading to composite modules and their algebraic connectors is illustrated by case studies. The essential novelty of this work resides in the minimal Occam’s razor set of algebraic connectors — another facet of Brooks’ Propriety principle leading to Conceptual Integrity of the whole Software System — within Linear Software Models, the unified algebraic theory of software modularity.

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Conjugate Laplacian matrices of a graph

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500829 ◽

2018 ◽

Vol 10 (06) ◽

pp. 1850082

Author(s):

Somnath Paul

Keyword(s):

Cartesian Product ◽

Laplacian Matrix ◽

Simple Graph ◽

Discrete Mathematics ◽

Laplacian Matrices ◽

Cartesian Product Of Graphs ◽

Laplacian Spectra ◽

The Matrix ◽

Matrix Formula ◽

Product Of Graphs

Let [Formula: see text] be a simple graph of order [Formula: see text] Let [Formula: see text] and [Formula: see text] where [Formula: see text] and [Formula: see text] are two nonzero integers and [Formula: see text] is a positive integer such that [Formula: see text] is not a perfect square. In [M. Lepovi[Formula: see text], On conjugate adjacency matrices of a graph, Discrete Mathematics 307 (2007) 730–738], the author defined the matrix [Formula: see text] to be the conjugate adjacency matrix of [Formula: see text] if [Formula: see text] for any two adjacent vertices [Formula: see text] and [Formula: see text] for any two nonadjacent vertices [Formula: see text] and [Formula: see text] and [Formula: see text] if [Formula: see text] In this paper, we define conjugate Laplacian matrix of graphs and describe various properties of its eigenvalues and eigenspaces. We also discuss the conjugate Laplacian spectra for union, join and Cartesian product of graphs.

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Small clique number graphs with three trivial critical ideals

Special Matrices ◽

10.1515/spma-2018-0011 ◽

2018 ◽

Vol 6 (1) ◽

pp. 122-154 ◽

Cited By ~ 5

Author(s):

Carlos A. Alfaro ◽

Carlos E. Valencia

Keyword(s):

Laplacian Matrix ◽

Clique Number ◽

Critical Group ◽

Characteristic Polynomials ◽

Laplacian Matrices ◽

Determinantal Ideals ◽

Invariant Factors ◽

Generalized Laplacian

Abstract The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. Previously, they have been used in the understanding and characterizing of the graphs with critical group with few invariant factors equal to one. However, critical ideals generalize the critical group, Smith group and the characteristic polynomials of the adjacency and Laplacian matrices of a graph. In this article we provide a set of minimal forbidden graphs for the set of graphs with at most three trivial critical ideals. Then we use these forbidden graphs to characterize the graphs with at most three trivial critical ideals and clique number equal to 2 and 3.

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Multipartite Separability of Laplacian Matrices of Graphs

The Electronic Journal of Combinatorics ◽

10.37236/150 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 3

Author(s):

Chai Wah Wu

Keyword(s):

Quantum Mechanics ◽

Physical Properties ◽

Laplacian Matrix ◽

Bipartite Graphs ◽

Density Matrices ◽

Laplacian Matrices ◽

Complete Bipartite Graphs ◽

Vertex Degrees ◽

Open Question ◽

Product Of Graphs

Recently, Braunstein et al. introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs whose normalized Laplacian matrix is multipartite entangled under any vertex labeling. Furthermore, we give conditions on the vertex degrees such that there is a vertex labeling under which the normalized Laplacian matrix is entangled. These results address an open question raised in Braunstein et al. Finally, we show that the Laplacian matrix of any product of graphs (strong, Cartesian, tensor, lexicographical, etc.) is multipartite separable, extending analogous results for bipartite and tripartite separability.

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Knot invariants from Laplacian matrices

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500585 ◽

2019 ◽

Vol 28 (09) ◽

pp. 1950058

Author(s):

Daniel S. Silver ◽

Susan G. Williams

Keyword(s):

Weighted Graph ◽

Laplacian Matrix ◽

Directed Edge ◽

Oriented Link ◽

Knot Invariants ◽

Laplacian Matrices ◽

Principal Submatrix ◽

Natural Way

A checkerboard graph of a special diagram of an oriented link is made a directed, edge-weighted graph in a natural way so that a principal submatrix of its Laplacian matrix is a Seifert matrix of the link. Doubling and weighting the edges of the graph produces a second Laplacian matrix such that a principal submatrix is an Alexander matrix of the link. The Goeritz matrix and signature invariants are obtained in a similar way. A device introduced by Kauffman makes it possible to apply the method to general diagrams.

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Spectral Clustering of Psychological Networks

10.31234/osf.io/fjqd4 ◽

2021 ◽

Author(s):

Michael Brusco ◽

Douglas Steinley ◽

Ashley L. Watts

Keyword(s):

Spectral Clustering ◽

Cluster Structure ◽

Laplacian Matrix ◽

Simulation Experiments ◽

Number Of Clusters ◽

Laplacian Matrices ◽

Undirected Network ◽

Close Relationship ◽

Recovery Performance ◽

Better Than

Spectral clustering is a well-known method for clustering the vertices of an undirected network. Although its use in network psychometrics has been limited, spectral clustering has a close relationship to the commonly-used walktrap algorithm. In this paper, we report results from four simulation experiments designed to evaluate the ability of spectral clustering and the walktrap algorithm to recover underlying cluster structure in networks. The salient findings include: (1) the cluster recovery performance of the walktrap algorithm can be improved slightly by using exact K-means clustering instead of hierarchical clustering; (2) K-means and K-median clustering led to comparable recovery performance when used to cluster vertices based on the eigenvectors of Laplacian matrices in spectral clustering; (3) spectral clustering using the unnormalized Laplacian matrix generally yielded inferior cluster recovery in comparison to the other methods; (4) when the correct number of clusters was provided for the methods, spectral clustering using the normalized Laplacian matrix led to better recovery than the walktrap algorithm; (5) when the number of clusters was unknown, spectral clustering using the normalized Laplacian matrix was appreciably better than the walktrap algorithm when the clusters were equally-sized, but the two methods were competitive when the clusters were not equally-sized. Overall, both the walktrap algorithm and spectral clustering of the normalized Laplacian matrix are effective for partitioning the vertices of undirected networks, with the latter performing better in most instances.

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Inertias of Laplacian matrices of weighted signed graphs

Special Matrices ◽

10.1515/spma-2019-0026 ◽

2019 ◽

Vol 7 (1) ◽

pp. 327-342

Author(s):

K. Hassani Monfared ◽

G. MacGillivray ◽

D. D. Olesky ◽

P. van den Driessche

Keyword(s):

Small Perturbation ◽

Infinite Family ◽

Laplacian Matrix ◽

Upper Bounds ◽

Signed Graph ◽

Signed Graphs ◽

Laplacian Matrices ◽

Combinatorial Parameter

Abstract We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights on the edges of any connected weighted signed graph so that all eigenvalues of its Laplacian matrix are simple. Next, we give upper bounds on the number of possible Laplacian inertias for signed graphs with a fixed flexibility τ (a combinatorial parameter of signed graphs), and show that these bounds are sharp for an infinite family of signed graphs. Finally, we provide upper bounds for the number of possible Laplacian inertias of signed graphs in terms of the number of vertices.

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Spectra of adjacency and Laplacian matrices of inhomogeneous Erdős–Rényi random graphs

Random Matrices Theory and Application ◽

10.1142/s201032632150009x ◽

2019 ◽

Vol 10 (01) ◽

pp. 2150009 ◽

Cited By ~ 2

Author(s):

Arijit Chakrabarty ◽

Rajat Subhra Hazra ◽

Frank den Hollander ◽

Matteo Sfragara

Keyword(s):

Social Networks ◽

Continuous Function ◽

Random Graphs ◽

Empirical Distribution ◽

Laplacian Matrix ◽

Laplacian Matrices ◽

Matrix Formula ◽

Spectral Distributions ◽

Special Case

This paper considers inhomogeneous Erdős–Rényi random graphs [Formula: see text] on [Formula: see text] vertices in the non-sparse non-dense regime. The edge between the pair of vertices [Formula: see text] is retained with probability [Formula: see text], [Formula: see text], independently of other edges, where [Formula: see text] is a continuous function such that [Formula: see text] for all [Formula: see text]. We study the empirical distribution of both the adjacency matrix [Formula: see text] and the Laplacian matrix [Formula: see text] associated with [Formula: see text], in the limit as [Formula: see text] when [Formula: see text] and [Formula: see text]. In particular, we show that the empirical spectral distributions of [Formula: see text] and [Formula: see text], after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where [Formula: see text] with [Formula: see text] a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, we apply our results to constrained random graphs, Chung–Lu random graphs and social networks.

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