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Characterization of fundamental networks

Part of: Graph theory

Published online by Cambridge University Press:  26 January 2019

Manuela A. D. Aguiar ,
Ana P. S. Dias  and
Pedro Soares
Manuela A. D. Aguiar
Affiliation:
Faculdade de Economia, Universidade do Porto, Rua Dr Roberto Frias, 4200-464 Porto, Portugal and Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007Porto, Portugal (maguiar@fep.up.pt)
Ana P. S. Dias
Affiliation:
Dep. Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007Porto, Portugal and Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal (apdias@fc.up.pt; ptcsoares@fc.up.pt)
Pedro Soares
Affiliation:
Dep. Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007Porto, Portugal and Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal (apdias@fc.up.pt; ptcsoares@fc.up.pt)
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Abstract

In the framework of coupled cell systems, a coupled cell network describes graphically the dynamical dependencies between individual dynamical systems, the cells. The fundamental network of a network reveals the hidden symmetries of that network. Subspaces defined by equalities of coordinates which are flow-invariant for any coupled cell system consistent with a network structure are called the network synchrony subspaces. Moreover, for every synchrony subspace, each network admissible system restricted to that subspace is a dynamical system consistent with a smaller network called a quotient network. We characterize networks such that: the network is a subnetwork of its fundamental network, and the network is a fundamental network. Moreover, we prove that the fundamental network construction preserves the quotient relation and it transforms the subnetwork relation into the quotient relation. The size of cycles in a network and the distance of a cell to a cycle are two important properties concerning the description of the network architecture. In this paper, we relate these two architectural properties in a network and its fundamental network.

Keywords

Fundamental networksnetwork connectivityrings

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary: 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 05C40: Connectivity
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 453 - 474
Copyright
Copyright © Royal Society of Edinburgh 2019

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