Applications of Graph Theory in Linear Algebra

1984 ◽  
Vol 57 (2) ◽  
pp. 67 ◽  
Author(s):  
Michael Doob
Keyword(s):  
Graph Theory ◽  
Linear Algebra

Applications of Graph Theory in Linear Algebra

1984 ◽  
Vol 57 (2) ◽  
pp. 67-76 ◽  
Author(s):  
Michael Doob
Keyword(s):  
Graph Theory ◽  
Linear Algebra

scholarly journals Balanced Centrality of Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Mark Debono ◽  
Josef Lauri ◽  
Irene Sciriha
Keyword(s):  
Graph Theory ◽  
Network Analysis ◽  
Linear Combination ◽  
Linear Algebra ◽  
Adjacency Matrix ◽  
Clear Understanding ◽  
Degree Centrality ◽  
Two Dimensional ◽  
Dimensional Array

There is an age-old question in all branches of network analysis. What makes an actor in a network important, courted, or sought? Both Crossley and Bonacich contend that rather than its intrinsic wealth or value, an actor’s status lies in the structures of its interactions with other actors. Since pairwise relation data in a network can be stored in a two-dimensional array or matrix, graph theory and linear algebra lend themselves as great tools to gauge the centrality (interpreted as importance, power, or popularity, depending on the purpose of the network) of each actor. We express known and new centralities in terms of only two matrices associated with the network. We show that derivations of these expressions can be handled exclusively through the main eigenvectors (not orthogonal to the all-one vector) associated with the adjacency matrix. We also propose a centrality vector (SWIPD) which is a linear combination of the square, walk, power, and degree centrality vectors with weightings of the various centralities depending on the purpose of the network. By comparing actors’ scores for various weightings, a clear understanding of which actors are most central is obtained. Moreover, for threshold networks, the (SWIPD) measure turns out to be independent of the weightings.

scholarly journals Spectrum Matriks Detour dari Graf Roda dengan n +1 Titik W_n

2020 ◽  
Vol 3 (1) ◽  
pp. 32
Author(s):  
Muhammad Abdy ◽  
Rahmat Syam ◽  
Agnes Monica Putri
Keyword(s):  
Graph Theory ◽  
Linear Algebra ◽  
Eigenvalues And Eigenvectors ◽  
Interesting Topic ◽  
Order Matrix ◽  
Wheel Graph

Penelitian ini bertujuan untuk  menentukan spectrum matriks detour dari graf roda dengan n+1 titik Wn. Spectrum dalam teori graf merupakan suatu topik menarik untuk dikaji dengan mempertemukan teori graf dan aljabar linear. Bentuk spectrum matriks detour adalah salah satu spectrum yang dapat ditentukan dalam graf roda. Matriks berordo (2 × n) yang terdiri dari nilai eigen berbeda dan banyak basis ruang eigen dari matriks terhubung langsung graf roda merupakan spectrum dari graf roda. Hasil penelitian ini menunjukkan bahwa langkah-langkah dalam menentukan spectrum matriks detour dari graf roda n+1 titik Wn, yaitu: menentukan graf roda dengan n + 1 titik Wn; menentukan detour, nilai eigen dan vektor eigen dari graf roda dengan n + 1 titik Wn,; melihat spectrum dan pola spectrum matriks detour dari graf roda n+1 titik Wn; pola yang didapat berupa dugaan kemudian dibuktikan dengan merumuskan suatu teorema yang dilengkapi dengan bukti.   Kata Kunci: Spectrum, Matriks Detour, Graf RodaThis study aims to determine the spectrum of detour matrix from the wheel graph with n+1 point Wn. Spectrum in graph theory is an interesting topic to review by bringing together graph theory and linear algebra. The form of the spectrum of detour matrix is one of the spectrums that can be determined in the wheel graph. The order matrix (2 × n) which consists of different eigenvalues and many the eigen space base from matrix adjacent wheel graph  is the spectrum of wheel graph. The results of this study show that steps in determining spectrum of detour matrix from the wheel graph with n+1 point Wn, that is: determine the wheel graph with n+1 point Wn; determine the detour; eigenvalues and eigenvectors of the wheel graph with n+1 point Wn; see the spectrum and patterns spectrum of detour matrix from the wheel graph with n+1 point Wn; pattern obtained in the form of conjecture then proved by formulating a theorem equipped with proof.Keywords: Spectrum, Detour Matrix, Wheel Graph.

Applications of Linear Algebra and Graph Theory

2017 ◽  
pp. 264-290
Author(s):  
Soren Eilers ◽  
Rune Johansen
Keyword(s):  
Graph Theory ◽  
Linear Algebra

scholarly journals A probability problem suitable for Problem-Based Learning

2021 ◽  
pp. 165-172
Author(s):  
Juan Luis González-Santander
Keyword(s):  
Graph Theory ◽  
Linear Algebra ◽  
Problem Based Learning ◽  
Undergraduate Level ◽  
Probability Problem ◽  
As Graph

We propose a simple probability problem for undergraduate level. This problem involves different branches of Mathematics, such as Graph Theory, Linear Algebra or hypergeometric sums, hence it is quite suitable to be used as Problem-Based Learning. In addition, the problem allows several variations so that it may be proposed to different groups of students at the same time.

Study on the Accessibility Analysis Method of Loop-Pipe-Network with Multi-Gas-Source

2011 ◽  
Vol 347-353 ◽  
pp. 3869-3872
Author(s):  
Wen Zhong Xu ◽  
Yan Hua Cheng ◽  
Wan Zheng Zhang
Keyword(s):  
Mathematical Model ◽  
Graph Theory ◽  
Linear Algebra ◽  
Solution Method ◽  
Analysis Method ◽  
Pipe Network ◽  
Accessibility Analysis ◽  
Gas Source ◽  
The Mathematical Model ◽  
Designing Method

This paper carries the study on the accessibility analysis method of loop-pipe-network with multi-gas-source. For the unreasonableness of conventional designing method of loop-pipe-network with multi-gas-source, the accessibility analysis mathematical model is established with relevant theory of graph theory, hydromechanics and linear algebra, and the solution method of the mathematical model of accessibility analysis is set. The accessibility analysis method could achieve the reliability and economy of network.

scholarly journals Pythagorean Fuzzy Matroids with Application

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 423 ◽  
Author(s):  
Muhammad Asif ◽  
Muhammad Akram ◽  
Ghous Ali
Keyword(s):  
Graph Theory ◽  
Fuzzy Sets ◽  
Linear Algebra ◽  
Vector Spaces ◽  
Algebraic Structures ◽  
Pythagorean Fuzzy Sets ◽  
Fuzzy Models ◽  
Rank Functions

The Pythagorean fuzzy models deal with graphical and algebraic structures in case of vague information related to membership and non-membership grades. Here, we use Pythagorean fuzzy sets to generalize the concept of vector spaces and discuss their basis and dimensions. We also highlight the concept of Pythagorean fuzzy matroids and examine some of their fundamental characteristics like circuits, basis, dimensions, and rank functions. Additionally, we explore the concept of Pythagorean fuzzy matroids in linear algebra, graph theory, and combinatorics. Finally, we demonstrate the use of Pythagorean fuzzy matroids for minimizing the time taken by a salesman in delivering given products.

Linear algebra methods in graph theory

2015 ◽  
Author(s):  
Elizaveta A. Kalinina ◽  
Gennady M. Khitrov ◽  
Sergey V. Pogozhev
Keyword(s):  
Graph Theory ◽  
Linear Algebra
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