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Keywords = nilpotent Cayley graphs

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29 pages, 3044 KB  
Article
Brauer Analysis of Some Cayley and Nilpotent Graphs and Its Application in Quantum Entanglement Theory
by Agustín Moreno Cañadas, Ismael Gutierrez and Odette M. Mendez
Symmetry 2024, 16(5), 570; https://doi.org/10.3390/sym16050570 - 6 May 2024
Cited by 7 | Viewed by 2534
Abstract
Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration [...] Read more.
Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages. Full article
(This article belongs to the Special Issue Symmetry in Graph Algorithms and Graph Theory III)
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80 pages, 903 KB  
Review
Graphs Defined on Rings: A Review
by S. Madhumitha and Sudev Naduvath
Mathematics 2023, 11(17), 3643; https://doi.org/10.3390/math11173643 - 23 Aug 2023
Cited by 9 | Viewed by 9943
Abstract
The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are [...] Read more.
The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area. Full article
(This article belongs to the Special Issue Advances in Topological Graph Theory)
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