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# Hyperbolicity on Graph Operators

by J. A. Méndez-Bermúdez 1 , Rosalío Reyes 2, José M. Rodríguez 2 and José M. Sigarreta 1,3,* 1
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
2
3
Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, Acapulco Gro. 39650, Mexico
*
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
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