scholarly journals Small clique number graphs with three trivial critical ideals

2018 ◽  
Vol 6 (1) ◽  
pp. 122-154 ◽  
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.

scholarly journals Properties of the characteristic polynomials and spectrum of Pn and Cn

2016 ◽  
Vol 5 (2) ◽  
pp. 132
Author(s):  
Essam El Seidy ◽  
Salah Eldin Hussein ◽  
Atef Mohamed
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Signless Laplacian Matrix ◽  
Path Graph ◽  
Signless Laplacian ◽  
Vertex Set ◽  
Cycle Graph

We consider a finite undirected and connected simple graph  with vertex set  and edge set .We calculated the general formulas of the spectra of a cycle graph and path graph. In this discussion we are interested in the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and seidel adjacency matrix.

scholarly journals EMPIRICAL DISTRIBUTIONS OF LAPLACIAN MATRICES OF LARGE DILUTE RANDOM GRAPHS

2012 ◽  
Vol 01 (03) ◽  
pp. 1250004 ◽  
Author(s):  
TIEFENG JIANG
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Spectral Properties ◽  
Empirical Distribution ◽  
Laplacian Matrix ◽  
Free Convolution ◽  
Laplacian Matrices ◽  
Empirical Distributions ◽  
Large N ◽  
Eigenvalues Of The Laplacian

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.

scholarly journals Computing the Permanent of the Laplacian Matrices of Nonbipartite Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Xiaoxue Hu ◽  
Grace Kalaso
Keyword(s):  
Laplacian Matrix ◽  
Laplacian Matrices ◽  
Recursion Formulas

Let G be a graph with Laplacian matrix L G . Denote by per  L G the permanent of L G . In this study, we investigate the problem of computing the permanent of the Laplacian matrix of nonbipartite graphs. We show that the permanent of the Laplacian matrix of some classes of nonbipartite graphs can be formulated as the composite of the determinants of two matrices related to those Laplacian matrices. In addition, some recursion formulas on per  L G are deduced.

scholarly journals Multi-View Spectral Clustering with Optimal Neighborhood Laplacian Matrix

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.

On the Laplacian characteristic polynomials of mixed graphs

2015 ◽  
Vol 30 ◽  
Author(s):  
Dariush Kiani ◽  
Maryam Mirzakhah
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Characteristic Polynomials ◽  
Laplacian Spectral Radius ◽  
Mixed Graph ◽  
First Derivative ◽  
Signless Laplacian ◽  
Mixed Graphs

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial of G are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic polynomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial of a mixed graph is reconstructible from the collection of the Laplacian characteristic polynomials of its edge deleted subgraphs. Then, it is investigated how graph modifications affect the mixed Laplacian characteristic polynomial. Also, a connection between the Laplacian characteristic polynomial of a non-singular connected mixed graph and the signless Laplacian characteristic polynomial is provided, and it is used to establish a lower bound for the spectral radius of L(G). Finally, using Coates digraphs, the perturbation of the mixed Laplacian spectral radius under some graph transformations is discussed.

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

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.

Conjugate Laplacian matrices of a graph

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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