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Mathematical Properties of the Hyperbolicity of Circulant Networks

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  • Juan C. Hernández
  • José M. Rodríguez
  • José M. Sigarreta

Abstract

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3], 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 the two other sides, for every geodesic triangle T in X. The study of the hyperbolicity constant in networks is usually a very difficult task; therefore, it is interesting to find bounds for particular classes of graphs. A network is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we obtain several sharp inequalities for the hyperbolicity constant of circulant networks; in some cases we characterize the graphs for which the equality is attained.

Suggested Citation

  • Juan C. Hernández & José M. Rodríguez & José M. Sigarreta, 2015. "Mathematical Properties of the Hyperbolicity of Circulant Networks," Advances in Mathematical Physics, John Wiley & Sons, vol. 2015(1).
    • Handle: RePEc:wly:jnlamp:v:2015:y:2015:i:1:n:723451
    DOI: 10.1155/2015/723451
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