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.

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 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 An Interesting Property of a Class of Circulant Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Seyed Morteza Mirafzal ◽  
Ali Zafari
Keyword(s):  
Adjacency Matrix ◽  
Cayley Graphs ◽  
Additive Group ◽  
Interesting Property ◽  
Circulant Graphs ◽  
Transitive Graph ◽  
Matrix Spectrum ◽  
Adjacency Matrix Spectrum

Suppose thatΠ=Cay(Zn,Ω)andΛ=Cay(Zn,Ψm)are two Cayley graphs on the cyclic additive groupZn, wherenis an even integer,m=n/2+1,Ω=t∈Zn∣t  is  odd, andΨm=Ω∪{n/2}are the inverse-closed subsets ofZn-0. In this paper, it is shown thatΠis a distance-transitive graph, and, by this fact, we determine the adjacency matrix spectrum ofΠ. Finally, we show that ifn≥8andn/2is an even integer, then the adjacency matrix spectrum ofΛisn/2+11,1-n/21,1n-4/2,-1n/2(we write multiplicities as exponents).

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.

Resilient Design of Complex Engineered Systems

2013 ◽  
Author(s):  
Hoda Mehrpouyan ◽  
Brandon Haley ◽  
Andy Dong ◽  
Irem Y. Tumer ◽  
Chris Hoyle
Keyword(s):  
Complex Network ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Graph Spectra ◽  
Complex Engineered Systems ◽  
Resilient Design ◽  
Spectra Properties ◽  
Key Driver ◽  
Engineered Systems

This paper presents a complex network and graph spectral approach to calculate the resiliency of complex engineered systems. Resiliency is a key driver in how systems are developed to operate in an unexpected operating environment, and how systems change and respond to the environments in which they operate. This paper deduces resiliency properties of complex engineered systems based on graph spectra calculated from their adjacency matrix representations, which describes the physical connections between components in a complex engineered systems. In conjunction with the adjacency matrix, the degree and Laplacian matrices also have eigenvalue and eigenspectrum properties that can be used to calculate the resiliency of the complex engineered system. One such property of the Laplacian matrix is the algebraic connectivity. The algebraic connectivity is defined as the second smallest eigenvalue of the Laplacian matrix and is proven to be directly related to the resiliency of a complex network. Our motivation in the present work is to calculate the algebraic connectivity and other graph spectra properties to predict the resiliency of the system under design.

scholarly journals Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

2014 ◽  
Vol 157 (1) ◽  
pp. 45-61
Author(s):  
PABLO SPIGA
Keyword(s):  
Automorphism Group ◽  
Positive Solution ◽  
Cayley Graph ◽  
Burnside Problem ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Restricted Burnside Problem ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractIn this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

scholarly journals Spectra of Graphs Resulting from Various Graph Operations and Products: a Survey

2018 ◽  
Vol 6 (1) ◽  
pp. 323-342 ◽  
Author(s):  
S. Barik ◽  
D. Kalita ◽  
S. Pati ◽  
G. Sahoo
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Explicit Description ◽  
Graph Products ◽  
Graph Operations ◽  
Signless Laplacian Matrix ◽  
Spectra Of Graphs ◽  
Signless Laplacian

AbstractLet G be a graph on n vertices and A(G), L(G), and |L|(G) be the adjacency matrix, Laplacian matrix and signless Laplacian matrix of G, respectively. The paper is essentially a survey of known results about the spectra of the adjacency, Laplacian and signless Laplacian matrix of graphs resulting from various graph operations with special emphasis on corona and graph products. In most cases, we have described the eigenvalues of the resulting graphs along with an explicit description of the structure of the corresponding eigenvectors.

scholarly journals On the Main Eigenvalues of Universal Adjacency Matrices and U-Controllable Graphs

2015 ◽  
Vol 30 ◽  
pp. 812-826
Author(s):  
Alexander Farrugia ◽  
Irene Sciriha
Keyword(s):  
Adjacency Matrix ◽  
Sufficient Conditions ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Identity Matrix ◽  
The Matrix ◽  
Vertex Degrees ◽  
Definition Of ◽  
Main Eigenvalues ◽  
Necessary And Sufficient

A universal adjacency matrix U of a graph G is a linear combination of the 0–1 adjacency matrix A, the diagonal matrix of vertex degrees D, the identity matrix I and the matrix J each of whose entries is 1. A main eigenvalue of U is an eigenvalue having an eigenvector that is not orthogonal to the all–ones vector. It is shown that the number of distinct main eigenvalues of U associated with a simple graph G is at most the number of orbits of any automorphism of G. The definition of a U–controllable graph is given using control–theoretic techniques and several necessary and sufficient conditions for a graph to be U–controllable are determined. It is then demonstrated that U–controllable graphs are asymmetric and that the converse is false, showing that there exist both regular and non–regular asymmetric graphs that are not U–controllable for any universal adjacency matrix U. To aid in the discovery of these counterexamples, a gamma–Laplacian matrix L(gamma) is used, which is a simplified form of U. It is proved that any U-controllable graph is a L(gamma)–controllable graph for some parameter gamma.

scholarly journals Laplacian integral graphs with a given degree sequence constraint

2021 ◽  
Vol 40 (6) ◽  
pp. 1431-1448
Author(s):  
Ansderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo de Lima
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

TWO SHARP UPPER BOUNDS FOR THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

2011 ◽  
Vol 03 (02) ◽  
pp. 185-191 ◽  
Author(s):  
YA-HONG CHEN ◽  
RONG-YING PAN ◽  
XIAO-DONG ZHANG
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The signless Laplacian matrix of a graph is the sum of its degree diagonal and adjacency matrices. In this paper, we present a sharp upper bound for the spectral radius of the adjacency matrix of a graph. Then this result and other known results are used to obtain two new sharp upper bounds for the signless Laplacian spectral radius. Moreover, the extremal graphs which attain an upper bound are characterized.

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