Jose Maria Sigarreta

On the hyperbolicity constant in graphs

If X is a geodesic metric space and x 1 , x 2 , x 3 ? X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics x 1 x 2 ] , x 2 x 3 ] and x 3 x 1 ] in X . The space X is ? -hyperbolic (in the Gromov sense) if, for every geodesic triangle T in X , every side of T is contained in a ? -neighborhood of the union of the other two sides. We denote by ? ( X ) the sharpest hyperbolicity constant of X , i.e. ? ( X ) ? inf { ? ? 0 : X ?is? ? -hyperbolic } . In this paper, we obtain several tight bounds for the hyperbolicity constant of a graph and precise values of this constant for some important families of graphs. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its number of edges, diameter and cycles. As a consequence of our results, we show that if G is any graph with m edges with lengths { l k } k = 1 m , then ? ( G ) ? ? k = 1 m l k / 4 , and ? ( G ) = ? k = 1 m l k / 4 if and only if G is isomorphic to C m . Moreover, we prove the inequality ? ( G ) ? 1 2 diam G for every graph, and we use this inequality in order to compute the precise value ? ( G ) for some common graphs.

Gromov hyperbolic graphs

Abstract In this paper we prove that the study of the hyperbolicity on graphs can be reduced to the study of the hyperbolicity on simpler graphs. In particular, we prove that the study of the hyperbolicity on a graph with loops and multiple edges can be reduced to the study of the hyperbolicity in the same graph without its loops and multiple edges; we also prove that the study of the hyperbolicity on an arbitrary graph is equivalent to the study of the hyperbolicity on a 3-regular graph obtained from it by adding some edges and vertices. Moreover, we study how the hyperbolicity constant of a graph changes upon adding or deleting finitely or infinitely many edges.

Gromov hyperbolic cubic graphs

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}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs.

Global offensive alliances in graphs

Abstract An offensive alliance in a graph Γ = ( V , E ) is a set of vertices S ⊂ V where for every vertex v in its boundary it holds that the majority of vertices in v s closed neighborhood are in S . In the case of strong offensive alliance, strict majority is required. An alliance S is called global if it affects every vertex in V \ S , that is, S is a dominating set of Γ. The global offensive alliance number γ o ( Γ ) (respectively, global strong offensive alliance number γ o ˆ ( Γ ) ) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in Γ. In this paper we obtain several tight bounds on γ o ( Γ ) and γ o ˆ ( Γ ) in terms of several parameters of Γ.

Spectral properties of geometric-arithmetic index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reflected in the algebraic properties of some matrices. The aim of this paper is to study the geometric-arithmetic index GA1 from an algebraic viewpoint. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices. Moreover, using this matrix, we define a GA Laplacian matrix which determines the geometric-arithmetic index of a graph and satisfies properties similar to the ones of the classical Laplacian matrix.

Small values of the hyperbolicity constant in graphs

If X is a geodesic metric space and x 1 , x 2 , x 3 ź X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics x 1 x 2 , x 2 x 3 and x 3 x 1 in X . The space X is ź -hyperbolic (in the Gromov sense) if any side of T is contained in a ź -neighborhood of the union of the two other sides, for every geodesic triangle T in X . We denote by ź ( X ) the sharpest hyperbolicity constant of X , i.e., ź ( X ) : = inf { ź ź 0 : X źisź ź -hyperbolic } . In the study of any parameter on graphs it is natural to study the graphs for which this parameter has small values. In this paper we study the graphs with small hyperbolicity constant, i.e., the graphs which are like trees (in the Gromov sense). We obtain simple characterizations of the graphs G with ź ( G ) = 1 and ź ( G ) = 5 4 (the case ź ( G ) < 1 is known). Also, we give a necessary condition in order to have ź ( G ) = 3 2 (we know that ź ( G ) is a multiple of 1 4 ). Although it is not possible to obtain bounds for the diameter of graphs with small hyperbolicity constant, we obtain such bounds for the effective diameter if ź ( G ) < 3 2 . This is the best possible result, since we prove that it is not possible to obtain similar bounds if ź ( G ) ź 3 2 .

On the complement graph and defensive k-alliances

In this paper, we obtain several tight bounds of the defensive k-alliance number in the complement graph from other parameters of the graph. In particular, we investigate the relationship between the alliance numbers of the complement graph and the minimum and maximum degree, the domination number and the isoperimetric number of the graph. Moreover, we prove the NP-completeness of the decision problem underlying the defensive k-alliance number.

Mathematical Properties of Gromov Hyperbolic Graphs

In this paper we deal with Gromov hyperbolic graphs. We obtain several tight bounds for the hyperbolicity constant of a graph. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its edge number. We prove that the study of the hyperbolicity on graphs can be reduced to the study of the hyperbolicity on simpler graphs. In particular, we show that the study of the hyperbolicity on a graph with loops and multiple edges can be reduced to the study of the hyperbolicity in its subjacent simple graph; we also prove that the study of the hyperbolicity of an arbitrary graph is equivalent to the study of the hyperbolicity of a 3‐regular graph obtained from it by adding some edges and vertices.

Partitioning a graph into offensive k -alliances

An offensive k -alliance in a graph is a set S of vertices with the property that every vertex in the boundary of S has at least k more neighbors in S than it has outside of S . An offensive k -alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) offensive k -alliances. The (global) offensive k -alliance partition number of a graph ? = ( V , E ) , denoted by ( ? k g o ( ? ) ) ? k o ( ? ) , is defined to be the maximum number of sets in a partition of V such that each set is an offensive (a global offensive) k -alliance. We show that 2 ? ? k g o ( ? ) ? 3 - k if ? is a graph without isolated vertices and k ? { 2 - Δ , . . . , 0 } . Moreover, we show that if ? is partitionable into global offensive k -alliances for k ? 1 , then ? k g o ( ? ) = 2 . As a consequence of the study we show that the chromatic number of ? is at most 3 if ? 0 g o ( ? ) = 3 . In addition, for k ? 0 , we obtain tight bounds on ? k o ( ? ) and ? k g o ( ? ) in terms of several parameters of the graph including the order, size, minimum and maximum degree. Finally, we study the particular case of the cartesian product of graphs, showing that ? k o ( ? 1 i? ? 2 ) ? ? k 1 o ( ? 1 ) ? k 2 o ( ? 2 ) , for k ? min { k 1 - Δ 2 , k 2 - Δ 1 } , where Δ i denotes the maximum degree of ? i , and ? k g o ( ? 1 i? ? 2 ) ? max { ? k 1 g o ( ? 1 ) , ? k 2 g o ( ? 2 ) } , for k ? min { k 1 , k 2 } .

On global offensive k-alliances in graphs

Abstract We investigate the relationship between global offensive k -alliances and some characteristic sets of a graph including r -dependent sets, τ -dominating sets and standard dominating sets. In addition, we discuss the close relationships that exist among the (global) offensive k i -alliance number of Γ i , i ∈ { 1 , 2 } , and the (global) offensive k -alliance number of Γ 1 × Γ 2 , for some specific values of k . As a consequence of the study, we obtain bounds on the global offensive k -alliance number in terms of several parameters of the graph.