scholarly journals Gromov Hyperbolicity in Strong Product Graphs

10.37236/3271 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Walter Carballosa ◽  
Rocío M. Casablanca ◽  
Amauris De la Cruz ◽  
José M. Rodríguez
Keyword(s):  
Metric Space ◽  
The Other ◽  
Gromov Hyperbolicity ◽  
Geodesic Triangle ◽  
Product Graph ◽  
Strong Product ◽  
Geodesic Metric Space ◽  
Product Graphs ◽  
Geodesic Metric ◽  
Hyperbolicity Constant

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta (X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta (X)=\inf\{\delta\geq 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}\,.$ In this paper we characterize the strong product of two graphs $G_1\boxtimes G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the strong product graph $G_1\boxtimes G_2$ is hyperbolic if and only if one of the factors is hyperbolic and the other one is bounded. We also prove some sharp relations between $\delta (G_1\boxtimes G_2)$, $\delta (G_1)$, $\delta (G_2)$ and the diameters of $G_1$ and $G_2$ (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the exact values of the hyperbolicity constant for many strong product graphs.

scholarly journals Distortion of the Hyperbolicity Constant of a Graph

10.37236/2175 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Walter Carballosa ◽  
Domingo Pestana ◽  
José M. Rodríguez ◽  
José M. Sigarreta
Keyword(s):  
Metric Space ◽  
Quantitative Information ◽  
The Other ◽  
Geodesic Triangle ◽  
Interesting Topic ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Hyperbolic Graphs ◽  
Two Sides ◽  
Hyperbolicity Constant

If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides, for every geodesic triangle $T$ in $X$. We denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta(X):=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims of this paper is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph $G\setminus e$ obtained from the graph $G$ by deleting an arbitrary edge $e$ from it. These inequalities allow to obtain the other main result of this paper, which characterizes in a quantitative way the hyperbolicity of any graph in terms of local hyperbolicity.

scholarly journals Mathematical Properties of the Hyperbolicity of Circulant Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Juan C. Hernández ◽  
José M. Rodríguez ◽  
José M. Sigarreta
Keyword(s):  
Cyclic Group ◽  
Metric Space ◽  
Geodesic Triangle ◽  
Group Of Automorphisms ◽  
Mathematical Properties ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Circulant Networks ◽  
Hyperbolicity Constant

IfXis a geodesic metric space andx1,x2,x3∈X, ageodesic triangle  T={x1,x2,x3}is the union of the three geodesics[x1x2],[x2x3], and[x3x1]inX. The spaceXisδ-hyperbolic(in the Gromov sense) if any side ofTis contained in aδ-neighborhood of the union of the two other sides, for every geodesic triangleTinX. 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.

scholarly journals On the Hyperbolicity Constant of Line Graphs

10.37236/697 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Walter Carballosa ◽  
José M. Rodríguez ◽  
José M. Sigarreta ◽  
María Villeta
Keyword(s):  
Metric Space ◽  
Line Graph ◽  
Line Graphs ◽  
Geodesic Triangle ◽  
Interesting Topic ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Quantitative Results ◽  
Hyperbolic Graphs ◽  
Hyperbolicity Constant

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. We denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta(X):=\inf\{\delta\ge 0: X \text{ is }\delta\text{-hyperbolic}\}$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. The main aim of this paper is to obtain information about the hyperbolicity constant of the line graph $\mathcal{L}(G)$ in terms of parameters of the graph $G$. In particular, we prove qualitative results as the following: a graph $G$ is hyperbolic if and only if $\mathcal{L}(G)$ is hyperbolic; if $\{G_n\}$ is a T-decomposition of $G$ ($\{G_n\}$ are simple subgraphs of $G$), the line graph $\mathcal{L}(G)$ is hyperbolic if and only if $\sup_n \delta(\mathcal{L}(G_n))$ is finite. Besides, we obtain quantitative results. Two of them are quantitative versions of our qualitative results. We also prove that $g(G)/4 \le \delta(\mathcal{L}(G)) \le c(G)/4+2$, where $g(G)$ is the girth of $G$ and $c(G)$ is its circumference. We show that $\delta(\mathcal{L}(G)) \ge \sup \{L(g):\, g \,\text{ is an isometric cycle in }\,G\,\}/4$. Furthermore, we characterize the graphs $G$ with $\delta(\mathcal{L}(G)) < 1$.

scholarly journals The hyperbolicity constant of infinite circulant graphs

2017 ◽  
Vol 15 (1) ◽  
pp. 800-814
Author(s):  
José M. Rodríguez ◽  
José M. Sigarreta
Keyword(s):  
Large Class ◽  
Cyclic Group ◽  
Metric Space ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Geodesic Triangle ◽  
Group Of Automorphisms ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Hyperbolicity Constant

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. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph 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 prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

scholarly journals On the hyperbolicity of edge-chordal and path-chordal graphs

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2599-2607 ◽  
Author(s):  
Sergio Bermudo ◽  
Walter Carballosa ◽  
José Rodríguez ◽  
José Sigarreta
Keyword(s):  
Chordal Graph ◽  
Chordal Graphs ◽  
Gromov Hyperbolicity ◽  
Geodesic Triangle ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Hyperbolic Graph ◽  
Hyperbolic Graphs ◽  
Definition Of ◽  
Two Sides

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 ?-hyperbolic (in the Gromov sense) if any side of T is contained in a ?-neighborhood of the union of the other two sides, for every geodesic triangle T in X. An important problem in the study of hyperbolic graphs is to relate the hyperbolicity with some classical properties in graph theory. In this paper we find a very close connection between hyperbolicity and chordality: we extend the classical definition of chordality in two ways, edge-chordality and path-chordality, in order to relate this propertywith Gromov hyperbolicity. In fact, we prove that every edge-chordal graph is hyperbolic and that every hyperbolic graph is path-chordal. Furthermore, we prove that every path-chordal cubic graph with small path-chordality constant is hyperbolic.

On the Hamilton Cycle of the Hypercube

2011 ◽  
Vol 480-481 ◽  
pp. 922-927 ◽  
Author(s):  
Yan Zhong Hu ◽  
Hua Dong Wang
Keyword(s):  
Euler Characteristic ◽  
Interconnection Networks ◽  
Cartesian Product ◽  
Hamilton Cycle ◽  
The Other ◽  
Product Graph ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Isomorphic Graphs ◽  
The Relationship

Hypercube is one of the basic types of interconnection networks. In this paper, we use the concept of the Cartesian product graph to define the hypercube Qn, we study the relationship between the isomorphic graphs and the Cartesian product graphs, and we get the result that there exists a Hamilton cycle in the hypercube Qn. Meanwhile, the other properties of the hypercube Qn, such as Euler characteristic and bipartite characteristic are also introduced.

scholarly journals Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Christopher H. Cashen
Keyword(s):  
Hyperbolic Space ◽  
Metric Space ◽  
Equivalence Classes ◽  
Product Topology ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Gromov Product ◽  
Geodesic Rays

AbstractWe consider a ‘contracting boundary’ of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space.We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example.

scholarly journals Growth tightness for groups with contracting elements

2014 ◽  
Vol 157 (2) ◽  
pp. 297-319 ◽  
Author(s):  
WEN-YUAN YANG
Keyword(s):  
Metric Space ◽  
Hyperbolic Group ◽  
Interesting Consequence ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Relatively Hyperbolic Group ◽  
Floyd Boundary ◽  
Relatively Hyperbolic

AbstractWe establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with non-trivial Floyd boundary are proven to be growth tight with respect to word metrics. In particular, all non-elementary relatively hyperbolic group are growth tight. This generalizes previous works of Arzhantseva-Lysenok and Sambusetti. Another interesting consequence is that CAT(0) groups with rank-1 elements are growth tight with respect to CAT(0)-metric.

scholarly journals On geometric polygroups

2020 ◽  
Vol 28 (1) ◽  
pp. 17-33
Author(s):  
F. Arabpur ◽  
M. Jafarpour ◽  
M. Aminizadeh ◽  
S. Hoskova-Mayerova
Keyword(s):  
Cayley Graph ◽  
Metric Space ◽  
Metric Spaces ◽  
Finitely Generated ◽  
Geodesic Metric Space ◽  
Geodesic Metric

AbstractIn this paper, we introduce a geodesic metric space called generalized Cayley graph (gCay(P,S)) on a finitely generated polygroup. We define a hyperaction of polygroup on gCayley graph and give some properties of this hyperaction. We show that gCayley graphs of a polygroup by two different generators are quasi-isometric. Finally, we express a connection between finitely generated polygroups and geodesic metric spaces.

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