Graph Partitioning via Adaptive Spectral Techniques

In this paper we study the use of spectral techniques for graph partitioning. Let G = (V, E) be a graph whose vertex set has a ‘latent’ partition V1,. . ., Vk. Moreover, consider a ‘density matrix’ Ɛ = (Ɛvw)v, sw∈V such that, for v ∈ Vi and w ∈ Vj, the entry Ɛvw is the fraction of all possible Vi−Vj-edges that are actually present in G. We show that on input (G, k) the partition V1,. . ., Vk can (very nearly) be recovered in polynomial time via spectral methods, provided that the following holds: Ɛ approximates the adjacency matrix of G in the operator norm, for vertices v ∈ Vi, w ∈ Vj ≠ Vi the corresponding column vectors Ɛv, Ɛw are separated, and G is sufficiently ‘regular’ with respect to the matrix Ɛ. This result in particular applies to sparse graphs with bounded average degree as n = #V → ∞, and it has various consequences on partitioning random graphs.

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Eigenpolytopes of Distance Regular Graphs

Canadian Journal of Mathematics ◽

10.4153/cjm-1998-040-8 ◽

1998 ◽

Vol 50 (4) ◽

pp. 739-755 ◽

Cited By ~ 4

Author(s):

C. D. Godsil

Keyword(s):

Convex Hull ◽

Orthogonal Projection ◽

Adjacency Matrix ◽

Gram Matrix ◽

Regular Graphs ◽

Largest Eigenvalue ◽

The Matrix ◽

Vertex Set ◽

Distance Regular Graphs ◽

Second Largest Eigenvalue

AbstractLet X be a graph with vertex set V and let A be its adjacency matrix. If E is the matrix representing orthogonal projection onto an eigenspace of A with dimension m, then E is positive semi-definite. Hence it is the Gram matrix of a set of |V| vectors in Rm. We call the convex hull of a such a set of vectors an eigenpolytope of X. The connection between the properties of this polytope and the graph is strongest when X is distance regular and, in this case, it is most natural to consider the eigenpolytope associated to the second largest eigenvalue of A. The main result of this paper is the characterisation of those distance regular graphs X for which the 1-skeleton of this eigenpolytope is isomorphic to X.

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Spectra of Edge-Independent Random Graphs

The Electronic Journal of Combinatorics ◽

10.37236/3576 ◽

2013 ◽

Vol 20 (4) ◽

Cited By ~ 9

Author(s):

Linyuan Lu ◽

Xing Peng

Keyword(s):

Random Graphs ◽

Adjacency Matrix ◽

Error Term ◽

Type Inequality ◽

Laplacian Matrix ◽

Mild Conditions ◽

Multiplicative Factor ◽

Indicator Variables ◽

Vertex Set ◽

General Graphs

Let $G$ be a random graph on the vertex set $\{1,2,\ldots, n\}$ such that edges in $G$ are determined by independent random indicator variables, while the probabilities $p_{ij}$ for $\{i,j\}$ being an edge in $G$ are not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of $G$ are recently studied by Oliveira and Chung-Radcliffe. Let $A$ be the adjacency matrix of $G$, $\bar A={\rm E}(A)$, and $\Delta$ be the maximum expected degree of $G$. Oliveira first proved that asymptotically almost surely $\|A-\bar A\|=O(\sqrt{\Delta \ln n})$ provided $\Delta\geq C \ln n$ for some constant $C$. Chung-Radcliffe improved the hidden constant in the error term using a new Chernoff-type inequality for random matrices. Here we prove that asymptotically almost surely $\|A-\bar A\|\leq (2+o(1))\sqrt{\Delta}$ with a slightly stronger condition $\Delta\gg \ln^4 n$. For the Laplacian matrix $L$ of $G$, Oliveira and Chung-Radcliffe proved similar results $\|L-\bar L\|=O(\sqrt{\ln n}/\sqrt{\delta})$ provided the minimum expected degree $\delta\geq C' \ln n$ for some constant $C'$; we also improve their results by removing the $\sqrt{\ln n}$ multiplicative factor from the error term under some mild conditions. Our results naturally apply to the classical Erdős–Rényi random graphs, random graphs with given expected degree sequences, and bond percolation of general graphs.

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Trees with Four and Five Distinct Signless Laplacian Eigenvalues

Journal of the Indonesian Mathematical Society ◽

10.22342/jims.25.3.557.302-313 ◽

2019 ◽

Vol 25 (3) ◽

pp. 302-313

Author(s):

Fatemeh Taghvaee ◽

Gholam Hossein Fath-Tabar

Keyword(s):

Adjacency Matrix ◽

Laplacian Matrix ◽

Simple Graph ◽

Laplacian Eigenvalues ◽

Signless Laplacian Matrix ◽

Signless Laplacian ◽

The Matrix ◽

Vertex Set

‎‎Let $G$ be a simple graph with vertex set $V(G)=\{v_1‎, ‎v_2‎, ‎\cdots‎, ‎v_n\}$ ‎and‎‎edge set $E(G)$‎.‎The signless Laplacian matrix of $G$ is the matrix $‎Q‎‎=‎D‎+‎A‎‎$‎, ‎such that $D$ is a diagonal ‎matrix‎%‎‎, ‎indexed by the vertex set of $G$ where‎‎%‎$D_{ii}$ is the degree of the vertex $v_i$ ‎‎‎ and $A$ is the adjacency matrix of $G$‎.‎%‎ where $A_{ij} = 1$ when there‎‎%‎‎is an edge from $i$ to $j$ in $G$ and $A_{ij} = 0$ otherwise‎.‎The eigenvalues of $Q$ is called the signless Laplacian eigenvalues of $G$ and denoted by $q_1$‎, ‎$q_2$‎, ‎$\cdots$‎, ‎$q_n$ in a graph with $n$ vertices‎.‎In this paper we characterize all trees with four and five distinct signless Laplacian ‎eigenvalues.‎‎‎

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Mathematics of networks

10.1093/oso/9780198805090.003.0006 ◽

2018 ◽

Author(s):

Mark Newman

Keyword(s):

Random Walks ◽

Adjacency Matrix ◽

Graph Partitioning ◽

Dynamic Networks ◽

Graph Laplacian ◽

Network Visualization ◽

Final Part ◽

Bipartite Networks ◽

Component Structure ◽

Basic Properties

An introduction to the mathematical tools used in the study of networks. Topics discussed include: the adjacency matrix; weighted, directed, acyclic, and bipartite networks; multilayer and dynamic networks; trees; planar networks. Some basic properties of networks are then discussed, including degrees, density and sparsity, paths on networks, component structure, and connectivity and cut sets. The final part of the chapter focuses on the graph Laplacian and its applications to network visualization, graph partitioning, the theory of random walks, and other problems.

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The normalized distance Laplacian

Special Matrices ◽

10.1515/spma-2020-0114 ◽

2021 ◽

Vol 9 (1) ◽

pp. 1-18

Author(s):

Carolyn Reinhart

Keyword(s):

Spectral Radius ◽

Characteristic Polynomial ◽

Adjacency Matrix ◽

Connected Graph ◽

Distance Matrix ◽

Laplacian Matrix ◽

Diagonal Matrix ◽

The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

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Spectral techniques applied to sparse random graphs

Random Structures and Algorithms ◽

10.1002/rsa.20089 ◽

2005 ◽

Vol 27 (2) ◽

pp. 251-275 ◽

Cited By ~ 65

Author(s):

Uriel Feige ◽

Eran Ofek

Keyword(s):

Random Graphs ◽

Spectral Techniques

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On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽

10.1007/s00453-020-00795-3 ◽

2021 ◽

Author(s):

Robert Ganian ◽

Sebastian Ordyniak ◽

M. S. Ramanujan

Keyword(s):

Polynomial Time ◽

Disjoint Paths ◽

Structural Parameter ◽

Input Graph ◽

Feedback Vertex Set ◽

Fpt Algorithms ◽

Edge Disjoint ◽

Fpt Algorithm ◽

Vertex Set ◽

Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

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The energy of integral circulant graphs with prime power order

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm110131003s ◽

2011 ◽

Vol 5 (1) ◽

pp. 22-36 ◽

Cited By ~ 21

Author(s):

J.W. Sander ◽

T. Sander

Keyword(s):

Adjacency Matrix ◽

Prime Power ◽

Prime Power Order ◽

Circulant Graph ◽

Closed Formula ◽

Circulant Graphs ◽

Vertex Set

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b ? Zn; gcd(a-b, n)? D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

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