scholarly journals Spectra of Edge-Independent Random Graphs

10.37236/3576 ◽  
2013 ◽  
Vol 20 (4) ◽  
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.

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 Graph Partitioning via Adaptive Spectral Techniques

2009 ◽  
Vol 19 (2) ◽  
pp. 227-284 ◽  
Author(s):  
AMIN COJA-OGHLAN
Keyword(s):  
Density Matrix ◽  
Random Graphs ◽  
Polynomial Time ◽  
Spectral Methods ◽  
Adjacency Matrix ◽  
Graph Partitioning ◽  
Operator Norm ◽  
Spectral Techniques ◽  
The Matrix ◽  
Vertex Set

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.

scholarly journals Trees with Four and Five Distinct Signless Laplacian Eigenvalues

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.‎‎‎

scholarly journals The normalized distance Laplacian

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.

scholarly journals The energy of integral circulant graphs with prime power order

2011 ◽  
Vol 5 (1) ◽  
pp. 22-36 ◽  
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.

scholarly journals Quasi-Random Graphs, Pseudo-Random Graphs and Pseudorandom Binary Sequences, I. (Quasi-Random Graphs)

2019 ◽  
Vol 14 (2) ◽  
pp. 103-126
Author(s):  
József Borbély ◽  
András Sárközy
Keyword(s):  
Random Graphs ◽  
Adjacency Matrix ◽  
Binary Sequence ◽  
Binary Sequences ◽  
Circulant Graph ◽  
Large Families

AbstractIn the last decades many results have been proved on pseudo-randomness of binary sequences. In this series our goal is to show that using many of these results one can also construct large families of quasi-random, pseudo-random and strongly pseudo-random graphs. Indeed, it will be proved that if the first row of the adjacency matrix of a circulant graph forms a binary sequence which possesses certain pseudorandom properties (and there are many large families of binary sequences known with these properties), then the graph is quasi-random, pseudo-random or strongly pseudo-random, respectively. In particular, here in Part I we will construct large families of quasi-random graphs along these lines. (In Parts II and III we will present and study constructions for pseudo-random and strongly pseudo-random graphs, respectively.)

scholarly journals Spectral property of certain class of graphs associated with generalized Bethe trees and transitive graphs

2008 ◽  
Vol 2 (2) ◽  
pp. 260-275 ◽  
Author(s):  
Yi-Zheng Fan ◽  
Li Shuang-Dong ◽  
Dong Liang
Keyword(s):  
Spectral Property ◽  
Adjacency Matrix ◽  
Rooted Tree ◽  
Laplacian Matrix ◽  
Transitive Graph ◽  
Tridiagonal Matrices ◽  
Equal Degree ◽  
Transitive Graphs ◽  
Structure Properties

A generalized Bethe tree is a rooted tree for which the vertices in each level having equal degree. Let Bk be a generalized Bethe tree of k level, and let T r be a connected transitive graph on r vertices. Then we obtain a graph Bk?T r from r copies of Bk and T r by appending r roots to the vertices of T r respectively. In this paper, we give a simple way to characterize the eigenvalues of the adjacency matrix A(Bk ? T r) and the Laplacian matrix L(Bk?T r) of Bk?T r by means of symmetric tridiagonal matrices of order k. We also present some structure properties of the Perron vectors of A(Bk?T r) and the Fiedler vectors of L(Bk ? T r). In addition, we obtain some results on transitive graphs.

On colouring random graphs

1975 ◽  
Vol 77 (2) ◽  
pp. 313-324 ◽  
Author(s):  
G. R. Grimmett ◽  
C. J. H. McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Subgraph ◽  
Vertex Set ◽  
Image Position

AbstractLet ωn denote a random graph with vertex set {1, 2, …, n}, such that each edge is present with a prescribed probability p, independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ωn is, with probability one,

scholarly journals Bounds of Laplacian Energy of a Hypercube Graph

2018 ◽  
Vol 7 (4.10) ◽  
pp. 582
Author(s):  
K. Ameenal Bibi ◽  
B. Vijayalakshmi ◽  
R. Jothilakshmi
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Positive Semidefinite ◽  
Laplacian Energy ◽  
Eigen Values ◽  
Degree Matrix

Let  Qn denote  the n – dimensional  hypercube  with  order   2n and  size n2n-1. The  Laplacian  L  is defined  by  L = D  where D is  the  degree  matrix and  A is  the  adjacency  matrix  with  zero  diagonal  entries.  The  Laplacian  is a  symmetric  positive  semidefinite.  Let  µ1 ≥ µ2 ≥ ....µn-1 ≥ µn = 0 be the eigen values of  the Laplacian matrix.  The  Laplacian  energy is defined as  LE(G) = . In  this  paper, we  defined  Laplacian  energy  of  a  Hypercube  graph  and  also attained  the  lower  bounds.   

scholarly journals Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yajing Wang ◽  
Yubin Gao
Keyword(s):  
Graph Theory ◽  
Spectral Radius ◽  
Adjacency Matrix ◽  
Upper Bounds ◽  
Simple Graph ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Spectral Graph ◽  
Vertex Set

Spectral graph theory plays an important role in engineering. Let G be a simple graph of order n with vertex set V=v1,v2,…,vn. For vi∈V, the degree of the vertex vi, denoted by di, is the number of the vertices adjacent to vi. The arithmetic-geometric adjacency matrix AagG of G is defined as the n×n matrix whose i,j entry is equal to di+dj/2didj if the vertices vi and vj are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus–Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.

Sign in / Sign up

Export Citation Format

Share Document