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 On the Spectrum of Laplacian Matrix

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Akbar Jahanbani ◽  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Matrix ◽  
Vertex Degrees ◽  
The Largest Eigenvalue

Let G be a simple graph of order n . The matrix ℒ G = D G − A G is called the Laplacian matrix of G , where D G and A G denote the diagonal matrix of vertex degrees and the adjacency matrix of G , respectively. Let l 1 G , l n − 1 G be the largest eigenvalue, the second smallest eigenvalue of ℒ G respectively, and λ 1 G be the largest eigenvalue of A G . In this paper, we will present sharp upper and lower bounds for l 1 G and l n − 1 G . Moreover, we investigate the relation between l 1 G and λ 1 G .

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.

scholarly journals Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Naglaa M. El-Shazly
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Identity Matrix ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Real Matrices ◽  
Definite Solution

In this paper necessary and sufficient conditions for the matrix equation to have a positive definite solution are derived, where , is an identity matrix, are nonsingular real matrices, and is an odd positive integer. These conditions are used to propose some properties on the matrices , . Moreover, relations between the solution and the matrices are derived.

Laplacian integral graphs with a given degreee sequence constraint

2021 ◽  
Author(s):  
Anderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo Silva de Lima
Keyword(s):  
Adjacency Matrix ◽  
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 is all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all L-integral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

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 On circulant codes with prescribed distances

1977 ◽  
Vol 16 (3) ◽  
pp. 361-369
Author(s):  
M. Deza ◽  
Peter Eades
Keyword(s):  
Sufficient Conditions ◽  
Difference Sets ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Weighing Matrices ◽  
Cyclic Difference Sets ◽  
The Matrix ◽  
Circulant Codes ◽  
Necessary And Sufficient ◽  
Square Matrix

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

scholarly journals Generalized Timelike Mannheim Curves in Minkowski Space-Time

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
M. Akyig~it ◽  
S. Ersoy ◽  
İ. Özgür ◽  
M. Tosun
Keyword(s):  
Minkowski Space ◽  
Sufficient Conditions ◽  
Space Time ◽  
Necessary And Sufficient Conditions ◽  
Minkowski Space Time ◽  
Mannheim Curve ◽  
Definition Of ◽  
Necessary And Sufficient

We give the definition of generalized timelike Mannheim curve in Minkowski space-time . The necessary and sufficient conditions for the generalized timelike Mannheim curve are obtained. We show some characterizations of generalized Mannheim curve.

Equivalent norms on Banach Jordan algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 261-270 ◽  
Author(s):  
M. A. Youngson
Keyword(s):  
Jordan Algebra ◽  
Sufficient Conditions ◽  
Equivalent Norm ◽  
Jordan Algebras ◽  
The Self ◽  
Equivalent Norms ◽  
Positive Elements ◽  
Definition Of ◽  
Necessary And Sufficient

1. Introduction. Recently Kaplansky suggested the definition of a suitable Jordan analogue of B*-algebras, which we call J B*-algebras (see (10) and (11)). In this article, we give a characterization of those complex unital Banach Jordan algebras which are J B*-algebras in an equivalent norm. This is done by generalizing results of Bonsall ((3) and (4)) to give necessary and sufficient conditions on a real unital Banach Jordan algebra under which it is the self-adjoint part of a J B*-algebra in an equivalent norm. As a corollary we also obtain a characterization of the cones in a Banach Jordan algebra which are the set of positive elements of a J B*-algebra.

The algebra of differential forms on a full matric bialgebra

1993 ◽  
Vol 114 (1) ◽  
pp. 111-130 ◽  
Author(s):  
A. Sudbery
Keyword(s):  
Vector Space ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Differential Forms ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Algebra Of Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Necessary And Sufficient

AbstractWe construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

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