scholarly journals Theoretical study of energy, inertia and nullity of phenylene and anthracene

2021 ◽  
Vol 19 (1) ◽  
pp. 541-547
Author(s):  
Zaheer Ahmad ◽  
Zeeshan Saleem Mufti ◽  
Muhammad Faisal Nadeem ◽  
Hani Shaker ◽  
Hafiz Muhammad Afzal Siddiqui
Keyword(s):  
Graph Theory ◽  
Adjacency Matrix ◽  
Theoretical Study ◽  
Characteristic Polynomials ◽  
Number Of Zeros ◽  
Negative Eigenvalues ◽  
The Absolute

Abstract Energy of a molecule plays an important role in physics, chemistry and biology. In mathematics, the concept of energy is used in graph theory to help other subjects such as chemistry and physics. In graph theory, nullity is the number of zeros extracted from the characteristic polynomials obtained from the adjacency matrix, and inertia represents the positive and negative eigenvalues of the adjacency matrix. Energy is the sum of the absolute eigenvalues of its adjacency matrix. In this study, the inertia, nullity and signature of the aforementioned structures have been discussed.

Digraph Energy of Directed Polygons

2020 ◽  
Vol 23 ◽  
Author(s):  
Bo Deng ◽  
Ning Yang ◽  
Weilin Liang ◽  
Xiaoyun Lu
Keyword(s):  
Molecular Orbital ◽  
Adjacency Matrix ◽  
Energy Levels ◽  
Order Relation ◽  
Theoretical Chemistry ◽  
Characteristic Polynomials ◽  
Molecular Graphs ◽  
Conjugated Hydrocarbons ◽  
Quasi Order ◽  
The Absolute

Background: The energy E(G) of G is defined as the sum the absolute values of the eigenvalues of its adjacency matrix. In theoretical chemistry, within the Hu ̈ckel molecular orbital (HMO) approximation, the energy levels of the π-electrons in molecules of conjugated hydrocarbons are related to the energy of the molecular graphs. Objective: Generally, the energy to digraphs was proposed. Methodology: Let Δ_n be the set consisting of digraphs with n vertices and each cycle having length≡2 mod(4). The set of all the n-order directed hollow k-polygons in Δ_n based on a k-polygon G is denoted by H_k (G). Results: In this research, by using the quasi-order relation over Δ_n and the characteristic polynomials of digraphs, we describe the directed hollow k-polygon with the maximum digraph energy in H_k (G). Conclusion: The n-order oriented hollow k-polygon with the maximum digraph energy among Hk(G) only contains a cycle. Moreover, such a cycle is the longest one produced in G.

Finite Characterization of the Coarsest Balanced Coloring of a Network

2020 ◽  
Vol 30 (14) ◽  
pp. 2050212
Author(s):  
Ian Stewart
Keyword(s):  
Graph Theory ◽  
Adjacency Matrix ◽  
Universal Cover ◽  
Running Time ◽  
Set Up ◽  
Balanced Coloring ◽  
Refinement Algorithm ◽  
Partition Refinement

Balanced colorings of networks correspond to flow-invariant synchrony spaces. It is known that the coarsest balanced coloring is equivalent to nodes having isomorphic infinite input trees, but this condition is not algorithmic. We provide an algorithmic characterization: two nodes have the same color for the coarsest balanced coloring if and only if their [Formula: see text]th input trees are isomorphic, where [Formula: see text] is the number of nodes. Here [Formula: see text] is the best possible. The proof is analogous to that of Leighton’s theorem in graph theory, using the universal cover of the network and the notion of a symbolic adjacency matrix to set up a partition refinement algorithm whose output is the coarsest balanced coloring. The running time of the algorithm is cubic in [Formula: see text].

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.

Binomial incidence matrix of a semigraph

2020 ◽  
pp. 2150017
Author(s):  
Jyoti Shetty ◽  
G. Sudhakara
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Systematic Approach ◽  
Line Graph ◽  
The Matrix ◽  
Theory Of Graphs

A semigraph, defined as a generalization of graph by  Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs. In this paper, we define a representing matrix of a semigraph [Formula: see text] and call it binomial incidence matrix of the semigraph [Formula: see text]. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on [Formula: see text] vertices can be obtained from the incidence matrix of the complete graph [Formula: see text].

scholarly journals Some New Results on Various Graph Energies of the Splitting Graph

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Zheng-Qing Chu ◽  
Saima Nazeer ◽  
Tariq Javed Zia ◽  
Imran Ahmed ◽  
Sana Shahid
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Absolute Value ◽  
Regular Splitting ◽  
The Absolute ◽  
Splitting Graph ◽  
Simple Connected Graph

The energy of a simple connected graph G is equal to the sum of the absolute value of eigenvalues of the graph G where the eigenvalue of a graph G is the eigenvalue of its adjacency matrix AG. Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graph S′G is a multiple of corresponding energy of a given graph G.

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.

An Algorithm for Determining the Position and Orientation of 3-D Objects from Images Using Spectral Graph Theory

2015 ◽  
Vol 770 ◽  
pp. 585-591
Author(s):  
Alexey Barinov ◽  
Aleksey Zakharov
Keyword(s):  
Graph Theory ◽  
Adjacency Matrix ◽  
Spectral Decomposition ◽  
Spectral Graph Theory ◽  
Feature Points ◽  
Image Comparison ◽  
Spectral Graph ◽  
Position And Orientation ◽  
Weighted Adjacency Matrix

This paper describes an algorithm for computing the position and orientation of 3-D objects by comparing graphs. The graphs are based on feature points of the image. Comparison is performed by a spectral decomposition with obtaining eigenvectors of weighted adjacency matrix of the graph.

Generation of Epicyclic Gear Trains of One Degree of Freedom

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Y. V. D. Rao ◽  
A. C. Rao
Keyword(s):  
Graph Theory ◽  
Adjacency Matrix ◽  
Kinematic Chain ◽  
Planetary Gear ◽  
Degree Of Freedom ◽  
Planetary Gear Trains ◽  
Epicyclic Gear ◽  
Gear Trains ◽  
Hamming Matrix

New planetary gear trains (PGTs) are generated using graph theory. A geared kinematic chain is converted to a graph and a graph in turn is algebraically represented by a vertex-vertex adjacency matrix. Checking for isomorphism needs to be an integral part of the enumeration process of PGTs. Hamming matrix is written from the adjacency matrix, using a set of rules, which is adequate to detect isomorphism in PGTs. The present work presents the twin objectives of testing for isomorphism and compactness using the Hamming matrices and moment matrices.

scholarly journals On the distribution of the zeros of the derivative of the Riemann zeta-function

2014 ◽  
Vol 157 (3) ◽  
pp. 425-442 ◽  
Author(s):  
STEPHEN LESTER
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Horizontal Distribution ◽  
Characteristic Polynomials ◽  
Unitary Matrices ◽  
Number Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Random Unitary Matrices

AbstractWe establish an asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For ℜ(s) = σ satisfying (log T)−1/3+ε ⩽ (2σ − 1) ⩽ (log log T)−2, we show that the number of zeros of ζ′(s) with imaginary part between zero and T and real part larger than σ is asymptotic to T/(2π(σ−1/2)) as T → ∞. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for σ in this range the zeros of ζ′(s) are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.

On the energy and Estrada index of Cayley graphs

2015 ◽  
Vol 07 (01) ◽  
pp. 1550005 ◽  
Author(s):  
Modjtaba Ghorbani
Keyword(s):  
Adjacency Matrix ◽  
Cayley Graphs ◽  
Estrada Index ◽  
The Absolute

The concept of energy of a graph was first defined in 1978 by Gutman as the sum of the absolute values of the eigenvalues of its adjacency matrix. Let λ1, λ2, …, λn be eigenvalues of graph Γ, then the Estrada index of Γ is defined as [Formula: see text]. The aim of this paper is to estimate the energy and Estrada index of Cayley graphs Cay (G, S) where G ≅ D2n, U6n and S is a normal symmetric generating subset of G.

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