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.

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 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 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 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 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 Cogrowth for group actions with strongly contracting elements

2018 ◽  
Vol 40 (7) ◽  
pp. 1738-1754 ◽  
Author(s):  
GOULNARA N. ARZHANTSEVA ◽  
CHRISTOPHER H. CASHEN
Keyword(s):  
Mapping Class Groups ◽  
Hyperbolic Spaces ◽  
Hyperbolic Groups ◽  
Mapping Class ◽  
Small Cancellation ◽  
Class Groups ◽  
Original Result ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Right Angled Artin Groups

Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$ and let $\unicode[STIX]{x1D6FF}_{N}$ and $\unicode[STIX]{x1D6FF}_{G}$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric, then $\unicode[STIX]{x1D6FF}_{N}/\unicode[STIX]{x1D6FF}_{G}>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk’s original result on free groups with respect to a word metric and a recent result of Matsuzaki, Yabuki and Jaerisch on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

Clique Partitions of Chordal Graphs

1993 ◽  
Vol 2 (4) ◽  
pp. 409-415 ◽  
Author(s):  
Paul Erdős ◽  
Edward T. Ordman ◽  
Yechezkel Zalcstein
Keyword(s):  
Chordal Graph ◽  
Chordal Graphs ◽  
Split Graph ◽  
Threshold Graph

To partition the edges of a chordal graph on n vertices into cliques may require as many as n2/6 cliques; there is an example requiring this many, which is also a threshold graph and a split graph. It is unknown whether this many cliques will always suffice. We are able to show that (1 − c)n2/4 cliques will suffice for some c > 0.

Christian-Muslim Dialogue

1988 ◽  
Vol 16 (3) ◽  
pp. 279-286
Author(s):  
George N. Malek
Keyword(s):  
Middle East ◽  
Critical Examination ◽  
Christian Mission ◽  
Future Mission ◽  
Psychological Examination ◽  
Western Concept ◽  
Economic Interaction ◽  
Definition Of ◽  
Two Sides ◽  
Theological Thinking

This article reviews the postwar development of the Western concept of Islam in light of the present conflicts between the Middle East and the West, and analyzes Christian mission through an historical, psychological examination of the motive of postwar Christian mission to the Middle East. It then presents the problem of Christian/Muslim relations in light of the fundamental issue facing the two sides, that is, religious misunderstanding, not political or economic interaction. The article then raises questions on the method and motive of postwar Christian mission to the Middle East, suggesting an alternative method for future mission. The paper takes the position that dialogue is the most productive form of contact between Christianity and Islam. It attempts to indicate, by critical examination, the potential points of tension, error, and reconciliation in the theological thinking of both. A major contribution of the paper is its affirmation and definition of a dialogue, its method and motive. Finally, the paper charts some solutions, theologically, psychologically, and cross-religiously.

Bonferroni-Type Inequalities via Chordal Graphs

2002 ◽  
Vol 11 (4) ◽  
pp. 349-351 ◽  
Author(s):  
KLAUS DOHMEN
Keyword(s):  
Type Inequality ◽  
Chordal Graph ◽  
Chordal Graphs ◽  
Finite Collection ◽  
Selection Of

Let {Av}v∈V be a finite collection of events and G = (V, E) be a chordal graph. Our main result – the chordal graph sieve – is a Bonferroni-type inequality where the selection of intersections in the estimates is determined by a chordal graph G. It interpolates between Boole's inequality (G empty) and the sieve formula (G complete). By varying G, several inequalities both well-known and new are obtained in a concise and unified way.

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.

Sign in / Sign up

Export Citation Format

Share Document