Bounds on Gromov Hyperbolicity Constant in Graphs

Proceedings Mathematical Sciences

José M Sigarreta¹  José M Rodríguez² 

[1] Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. Col. Garita, 0 Acalpulco Gro., Mexico$$;Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 0, Leganés, Madrid, Spain$$

关键词: Infinite graphs;    Cartesian product graphs;    independence number;    domination number;    geodesics;    Gromov hyperbolicity.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

If 𝑋 is a geodesic metric space and 𝑥1,𝑥2,𝑥3 $in$ 𝑋, a geodesic triangle 𝑇={𝑥1,𝑥2,𝑥3} is the union of the three geodesics [𝑥1,𝑥2], [𝑥2,𝑥3] and [𝑥3𝑥1] in 𝑋. The space 𝑋 is 𝛿-hyperbolic (in the Gromov sense) if any side of 𝑇 is contained in a 𝛿-neighborhood of the union of two other sides, for every geodesic triangle 𝑇 in 𝑋. If 𝑋 is hyperbolic, we denote by 𝛿(𝑋) the sharp hyperbolicity constant of 𝑋, i.e. 𝛿(𝑋)=$inf{$𝛿≥ 0$ : 𝑋 is 𝛿-hyperbolic}. In this paper we relate the hyperbolicity constant of a graph with some known parameters of the graph, as its independence number, its maximum and minimum degree and its domination number. Furthermore, we compute explicitly the hyperbolicity constant of some class of product graphs.

【 授权许可】

Unknown