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Gromov hyperbolicity in lexicographic product graphs.
- Source :
-
Proceedings of the Indian Academy of Sciences: Mathematical Sciences . Feb2019, Vol. 129 Issue 1, p1-1. 1p. - Publication Year :
- 2019
-
Abstract
- If X is a geodesic metric space and x1,x2,x3∈X, a geodesic triangleT={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. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X)=inf{δ≥0:Xisδ-hyperbolic}. In this paper, we characterize the lexicographic product of two graphs G1∘G2 which are hyperbolic, in terms of G1 and G2: the lexicographic product graph G1∘G2 is hyperbolic if and only if G1 is hyperbolic, unless if G1 is a trivial graph (the graph with a single vertex); if G1 is trivial, then G1∘G2 is hyperbolic if and only if G2 is hyperbolic. In particular, we obtain the sharp inequalities δ(G1)≤δ(G1∘G2)≤δ(G1)+3/2 if G1 is not a trivial graph, and we characterize the graphs for which the second inequality is attained. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 02534142
- Volume :
- 129
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Proceedings of the Indian Academy of Sciences: Mathematical Sciences
- Publication Type :
- Academic Journal
- Accession number :
- 134310229
- Full Text :
- https://doi.org/10.1007/s12044-018-0451-y