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Open AccessArticle

Hyperbolicity on Graph Operators

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Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
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Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain
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Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, Acapulco Gro. 39650, Mexico
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Author to whom correspondence should be addressed.
Symmetry 2018, 10(9), 360; https://doi.org/10.3390/sym10090360
Received: 24 July 2018 / Revised: 16 August 2018 / Accepted: 22 August 2018 / Published: 24 August 2018
(This article belongs to the Special Issue Symmetry in Graph Theory)
A graph operator is a mapping F : Γ Γ , where Γ and Γ are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Λ ( G ) ; subdivision graph, S ( G ) ; total graph, T ( G ) ; and the operators R ( G ) and Q ( G ) . In particular, we get relationships such as δ ( G ) δ ( R ( G ) ) δ ( G ) + 1 / 2 , δ ( Λ ( G ) ) δ ( Q ( G ) ) δ ( Λ ( G ) ) + 1 / 2 , δ ( S ( G ) ) 2 δ ( R ( G ) ) δ ( S ( G ) ) + 1 and δ ( R ( G ) ) 1 / 2 δ ( Λ ( G ) ) 5 δ ( R ( G ) ) + 5 / 2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices. View Full-Text
Keywords: graph operators; gromov hyperbolicity; geodesics graph operators; gromov hyperbolicity; geodesics
MDPI and ACS Style

Méndez-Bermúdez, J.A.; Reyes, R.; Rodríguez, J.M.; Sigarreta, J.M. Hyperbolicity on Graph Operators. Symmetry 2018, 10, 360.

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