Basic Graph Theory and Graphical Models

Author(s):  
Russell G. Almond ◽  
Robert J. Mislevy ◽  
Linda S. Steinberg ◽  
Duanli Yan ◽  
David M. Williamson
1998 ◽  
Vol 09 (01) ◽  
pp. 25-37 ◽  
Author(s):  
THOMAS J. CORTINA ◽  
ZHIWEI XU

We present a family of interconnection networks named the Cube-Of-Rings (COR) networks along with their basic graph-theoretic properties. Aspects of group graph theory are used to show the COR networks are symmetric and optimally fault tolerant. We present a closed-form expression of the diameter and optimal one-to-one routing algorithm for any member of the COR family. We also discuss the suitability of the COR networks as the interconnection network of scalable parallel computers.


2021 ◽  
Vol 14 (3) ◽  
pp. 78
Author(s):  
Thomas Konstantinovsky ◽  
Matan Mizrachi

We propose a new approach to text semantic analysis and general corpus analysis using, as termed in this article, a "bi-gram graph" representation of a corpus. The different attributes derived from graph theory are measured and analyzed as unique insights or against other corpus graphs, attributes such as the graph chromatic number and the graph coloring, graph density and graph K-core. We observe a vast domain of tools and algorithms that can be developed on top of the graph representation; creating such a graph proves to be computationally cheap, and much of the heavy lifting is achieved via basic graph calculations. Furthermore, we showcase the different use-cases for the bi-gram graphs and how scalable it proves to be when dealing with large datasets.


Author(s):  
Z. Zhyvko ◽  
O. Voronka ◽  
V. Blikhar ◽  
I. Ruda ◽  
L. Kukharska

Abstract. The conducted theoretical and analytical study, which included acquaintance with the achievements of domestic and foreign scientists on ensuring the personnel security of banks and assessing the personnel security of the banking sector of the economy, allowed to develop oriented graphical models and corresponding reachability matrices, and served as a basis for hierarchical ordering of key threats to bank personnel security. The hierarchical ordering of threats to the personnel security of the banking sector of the economy is based on the application of the graph theory method. The models developed give an idea of the need to implement safeguards against key threats, which is important in view of resource constraints and maximizing the effectiveness of security actors. The result is averaged for most domestic banks, as each of them is influenced by the external environment and changes caused by internal processes. A mechanism for counteracting the influence of external and internal threats on the personnel security of banks has been constructed, which takes into account the current trends in the development of the banking sector. The application of system analysis and graph theory made it possible to hierarchically regulate the impact of key threats on personnel security, which became the basis for the development of a model that graphically determines the priority of developing and implementing security measures for each threat, and justifies the possibility of applying one of the options passive protection and active protection. As a result of the theoretical and analytical study, a number of key external and internal threats to personnel security of banks have been regulated and identified. By applying graph theory and mathematical modeling hierarchically ordered and determined the severity of the impact of threats on the personnel security of banks, which subsequently allowed to create a defense mechanism to counter them. Keywords: bank personnel security, bank system, threat, security, crisis situation, mechanism. JEL Classіfіcatіon D81, G21, F29, L20 Formulas: 16; fig.: 3; tabl.: 19; bibl.: 16.


10.37236/9446 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Shane Chern

In this paper, we start by considering generating function identities for linked partition ideals in the setting of basic graph theory. Then our attention is turned to $q$-difference systems, which eventually lead to a factorization problem of a special type of column functional vectors involving $q$-multi-summations. Using a recurrence relation satisfied by certain $q$-multi-summations, we are able to provide non-computer-assisted proofs of some Andrews--Gordon type generating function identities. These proofs also have an interesting connection with binary trees. Further, we give illustrations of constructing a linked partition ideal, or more loosely, a set of integer partitions whose generating function corresponds to a given set of special $q$-multi-summations.


Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 919
Author(s):  
Petr Coufal ◽  
Štěpán Hubálovský ◽  
Marie Hubálovská

Discrete mathematics covers the field of graph theory, which solves various problems in graphs using algorithms, such as coloring graphs. Part of graph theory is focused on algorithms that solve the passage through mazes and labyrinths. This paper presents a study conducted as part of a university course focused on graph theory. The course addressed the problem of high student failure in the mazes and labyrinths chapter. Students’ theoretical knowledge and practical skills in solving algorithms in the maze were low. Therefore, the use of educational robots and their involvement in the teaching of subjects in part focused on mazes and labyrinths. This study shows an easy passage through the individual areas of teaching the science, technology, engineering, and mathematics (STEM) concept. In this article, we describe the research survey and focus on the description and examples of teaching in a university course. Part of the work is the introduction of an easy transition from the theoretical solution of algorithms to their practical implementation on a real autonomous robot. The theoretical part of the course introduced the issues of graph theory and basic algorithms for solving the passage through the labyrinth. The contribution of this study is a change in the approach to teaching graph theory and a greater interconnection of individual areas of STEM to achieve better learning outcomes for science students.


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