On the Hyperbolicity Constant of Line Graphs

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. We denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X) := inf{δ ≥ 0 : X is δ-hyperbolic } .

On the hyperbolicity constant of circular-arc graphs

“…. , u n , u 1 and by C 0 its corresponding cycle in G. The following result in [11,Corollary 3.12] relates the hyperbolicity constants of G and L(G). If we consider the four-point definition of hyperbolicity, another usual definition, and we denote by δ ′ the sharp constant for this definition, we have the following result in [18,Theorem 6].…”

Gromov hyperbolicity in lexicographic product graphs

“…Therefore, it is interesting to study the hyperbolicity of particular classes of graphs. The papers [5,6,7,10,11,27,31,32,34,38,39] study the hyperbolicity of, respectively, complement of graphs, chordal graphs, strong product graphs, corona and join of graphs, line graphs, Cartesian product graphs, cubic graphs, tessellation graphs, short graphs, median graphs and k-chordal graphs. In [7,10,27] the authors characterize the hyperbolic product graphs (for strong product, corona and join of graphs, and Cartesian product) in terms of properties of their factor graphs.…”

“…The study of Gromov hyperbolic graphs is a subject of increasing interest in graph theory; see, e.g., [2,3,4,5,6,7,9,11,15,20,21,22,24,26,27,29,30,31,32,37,39] and the references therein.…”

“…Therefore, it is interesting to study the invariance of the hyperbolicity of graphs under appropriate transformations. The invariance of the hyperbolicity under some natural transformations on graphs have been studied in previous papers, for instance, removing edges of a graph is studied in [4,7], the line graph of a graph in [9,11], the dual of a planar graph in [8,29] and the complement of a graph in [5].…”

Bounds on Gromov hyperbolicity constant

“…The study of mathematical properties of Gromov hyperbolic spaces and its applications is a topic of recent and increasing interest in graph theory; see, for instance, [7,12,14,16,18,23,32,33,34,35,38,39,44,47,49].…”