scholarly journals On Tosha-degree of an Edge in a Graph

2020 ◽  
Vol 13 (5) ◽  
pp. 1097-1109
Author(s):  
R. Rajendra ◽  
P. Siva Kota Reddy
Keyword(s):  
Adjacency Matrix ◽  
Line Graph ◽  
Multiple Edges ◽  
Definition Of

In an earlier paper,  we have introduced the Tosha-degree of an edge in a graph without multiple edges and studied some properties. In this paper, we extend the definition of  Tosha-degree of an edge in a graph in which multiple edges are allowed. Also, we introduce the concepts - zero edges in a graph, $T$-line graph of a multigraph, Tosha-adjacency matrix, Tosha-energy, edge-adjacency matrix and edge energy of a graph $G$ and obtain some results.

Extending Proximity Measures to Attributed Networks for Community Detection

2021 ◽  
Vol 30 (4) ◽  
pp. 441-455
Author(s):  
Rinat Aynulin ◽  
◽  
Pavel Chebotarev ◽  
◽  
Keyword(s):  
Network Analysis ◽  
Community Detection ◽  
Network Structure ◽  
Real World ◽  
Adjacency Matrix ◽  
Attribute Information ◽  
Node Attributes ◽  
Proximity Measures ◽  
Definition Of ◽  
Attributed Networks

Proximity measures on graphs are extensively used for solving various problems in network analysis, including community detection. Previous studies have considered proximity measures mainly for networks without attributes. However, attribute information, node attributes in particular, allows a more in-depth exploration of the network structure. This paper extends the definition of a number of proximity measures to the case of attributed networks. To take node attributes into account, attribute similarity is embedded into the adjacency matrix. Obtained attribute-aware proximity measures are numerically studied in the context of community detection in real-world networks.

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 Quantum Symmetries of Graph C*-algebras

2018 ◽  
Vol 61 (4) ◽  
pp. 848-864 ◽  
Author(s):  
Simon Schmidt ◽  
Moritz Weber
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Operator Algebras ◽  
Automorphism Groups ◽  
Undirected Graphs ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Multiple Edges ◽  
Definition Of

AbstractThe study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

scholarly journals How the Statistical Validation of Functional Connectivity Patterns Can Prevent Erroneous Definition of Small-World Properties of a Brain Connectivity Network

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
J. Toppi ◽  
F. De Vico Fallani ◽  
G. Vecchiato ◽  
A. G. Maglione ◽  
F. Cincotti ◽  
...  
Keyword(s):  
Functional Connectivity ◽  
Adjacency Matrix ◽  
Brain Connectivity ◽  
Small World ◽  
Connectivity Pattern ◽  
Edge Density ◽  
Statistical Validation ◽  
Common Procedure ◽  
Connectivity Patterns ◽  
Definition Of

The application of Graph Theory to the brain connectivity patterns obtained from the analysis of neuroelectrical signals has provided an important step to the interpretation and statistical analysis of such functional networks. The properties of a network are derived from the adjacency matrix describing a connectivity pattern obtained by one of the available functional connectivity methods. However, no common procedure is currently applied for extracting the adjacency matrix from a connectivity pattern. To understand how the topographical properties of a network inferred by means of graph indices can be affected by this procedure, we compared one of the methods extensively used in Neuroscience applications (i.e. fixing the edge density) with an approach based on the statistical validation of achieved connectivity patterns. The comparison was performed on the basis of simulated data and of signals acquired on a polystyrene head used as a phantom. The results showed (i) the importance of the assessing process in discarding the occurrence of spurious links and in the definition of the real topographical properties of the network, and (ii) a dependence of the small world properties obtained for the phantom networks from the spatial correlation of the neighboring electrodes.

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 Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Harishchandra S. Ramane ◽  
Shaila B. Gudimani ◽  
Sumedha S. Shinde
Keyword(s):  
Characteristic Polynomial ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Line Graph ◽  
Induced Subgraphs ◽  
Subdivision Graph ◽  
Signless Laplacian ◽  
The Matrix ◽  
Derivatives Of ◽  
Degree Matrix

The signless Laplacian polynomial of a graph G is the characteristic polynomial of the matrix Q(G)=D(G)+A(G), where D(G) is the diagonal degree matrix and A(G) is the adjacency matrix of G. In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs.

scholarly journals Minimal line graphs

1976 ◽  
Vol 17 (1) ◽  
pp. 12-16 ◽  
Author(s):  
David P. Sumner
Keyword(s):  
Line Graph ◽  
Minimal Degree ◽  
Line Graphs ◽  
Multiple Edges

In this paper all graphs will be ordinary graphs, i.e. finite, undirected, and without loops or multiple edges. For points x and y of a graph G, we shall indicate that x is adjacent to y by writing x ⊥ y, and if x is not adjacent to y we shall write xy. We shall denote the degree of a point x by δ(x) and the minimal degree of G by δ(G).By the line graph of a graph G we shall mean the graph L(G) whose points are the edges of G, with two points of L(G) adjacent whenever they are adjacent in G. A graph G is said to be a line graph if there exists a graph H such that G = L(H).

scholarly journals Cognitive Modeling of Socio-Economic Systems with Functional Consideration of the Time Factor

2021 ◽  
Vol 93 ◽  
pp. 03022
Author(s):  
Aleksey Rogachev ◽  
Elena Melikhova
Keyword(s):  
Adjacency Matrix ◽  
Cognitive Modeling ◽  
Mutual Influence ◽  
Control Function ◽  
Parametric Identification ◽  
Cognitive Models ◽  
Time Factor ◽  
Exponential Dependence ◽  
Functional Dependencies ◽  
Definition Of

The research analyzes scientific and methodological approaches for structural and parametric identification of fuzzy cognitive models based on time-dependent functional dependencies. In the process of constructing and identifying fuzzy cognitive models, the stages of structural identification with the definition of a set of concepts and unclear relationships over this set, parametric identification, which implements the transition to fuzzy identification with the definition of the intensity of influence between factors, are implemented. A specialized software system was developed for the purpose of computer support for modeling and studying the influence of the time factor set by changes in the strength of connections between concepts. A representation of each of the elements of the FCM adjacency matrix is proposed in the form of an additive expression controlled by the alpha parameter, which includes a constant component determined by the expert method and a control function that depends on real or dimensionless model time. To correct the elements of the FCM adjacency matrix, it is proposed to use a 3-parameter parabolic or modified 4-parameter exponential dependence as a function that depends on real or dimensionless model time. The developed modified method for solving the problem of time factor accounting using functional dependencies for the strength of mutual influence of concepts provides for expanding the capabilities of fuzzy cognitive models by taking into account the time factor.

scholarly journals On the nullity of connected graphs with least eigenvalue at least -2

2013 ◽  
Vol 7 (2) ◽  
pp. 250-261 ◽  
Author(s):  
Jiang Zhou ◽  
Lizhu Sun ◽  
Hongmei Yao ◽  
Changjiang Bu
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Line Graph ◽  
Line Graphs ◽  
Connected Graphs ◽  
Least Eigenvalue

Let L (resp. L+) be the set of connected graphs with least adjacency eigenvalue at least -2 (resp. larger than -2). The nullity of a graph G, denoted by ?(G), is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. In this paper, we give the nullity set of L+ and an upper bound on the nullity of exceptional graphs. An expression for the nullity of generalized line graphs is given. For G ? L, if ?(G) is sufficiently large, then G is a proper generalized line graph (G is not a line graph).

The Mathematical Definition

2018 ◽  
Author(s):  
Ginestra Bianconi
Keyword(s):  
Adjacency Matrix ◽  
Fundamental Importance ◽  
Matrix Formalism ◽  
Multilayer Networks ◽  
Multilayer Network ◽  
Full Generality ◽  
Multiplex Networks ◽  
Mathematical Definition ◽  
Definition Of

This chapter provides the mathematical definition of multilayer networks and it is of fundamental importance for the rest of the book. The mathematical definition of multilayer networks is given in full generality and subsequently applied to specific types of multilayer networks including multiplex networks, multi-slice networks and networks of networks, motivating the discussion with examples and applications. The fundamental terminology used in multilayer networks is here introduced, including replica nodes, supernetworks and the supra-adjacency matrix. Additionally, this chapter describes the most efficient way to store a multilayer network dataset using a Multilayer Network Edgelist. Although this chapter focuses mostly on a matrix formalism to describe multilayer networks, a paragraph is devoted to the tensorial formalism for studying multilayer networks.

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