On the Geometric-Arithmetic Index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index GA1 and characterize graphs extremal with respect to them. In particular, we improve some known inequalities and we relate GA1 to other well known topological indices.Publicad

New results on the harmonic index and its generalizations

In this paper we obtain new inequalities involving the harmonic index and the(general) sum-connectivity index, and characterize graphs extremal with respect tothem. In particular, we improve and generalize some known inequalities and werelate this indices to other well-known topological indices.The authors are grateful to the referees for their valuable comments which have improved this paper. This work is supported in part by two grants from Ministerio de Economía y Competititvidad (MTM2013-46374-P and MTM2015-69323-REDT), Spain, and a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México

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 GA(1) 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. (C) 2015 Elsevier Inc. All rights reserved.This research was supported in part by a Grant from Ministerio de Economía y Competitividad (MTM 2013-46374-P), Spain, and a Grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México

Alianzas en grafos

En este trabajo estudiamos propiedades matemáticas de las k-alianzas en grafos y prestamos especial interés a la relación que existe entre el número de k-alianza (defensiva, ofensiva y dual) y otros parámetros conocidos como, por ejemplo, el orden, la medida, el cuello, el diámetro, el número de independencia, el número de dominación, la conectividad algebraica y el radio espectral. En algunos casos obtenemos el valor exacto del número de k-alianza y, en general, obtenemos cotas tensas no triviales para dicho parámetro. En el caso del grafo línea, se obtienen resultados sobre el número de alianza (defensiva y ofensiva) en función de parámetros conocidos del grafo original. A lo largo de toda la memoria particularizamos al caso de grafos planares y de grafos cúbicos. Estudiamos, además, la relación entre alianzas defensivas y ofensivas, así como las principales propiedades de los conjuntos libres de k-alianzas y de los cubrimientos de k-alianzas. Otra de las aportaciones de esta memoria es el inicio del estudio de las k-alianzas conexas y de las k-alianzas independientes, as´ı como el estudio de la relación entre los conjuntos k-dominantes totales y las k-alianzas (defensivas, ofensivas y duales). Esta memoria está estructurada en tres capítulos. Los dos primeros, aunque de similar estructura, son independientes y están dedicados al estudio de las k-alianzas defensivas y de las k-alianzas ofensivas, respectivamente. En el Capítulo 3 estudiamos las k-alianzas duales, los conjuntos k-dominantes totales, así como los cubrimientos y los conjuntos libres de k-alianzas

Optimal upper bounds of the geometric-arithmetic index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new upper bounds of the geometric-arithmetic index and characterize graphs extremal with respect to them.This research was supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain

Spectral study of the 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 reected 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.Supported in part by a grant from Ministerio de Economía y Competititvidad (MTM 2013-46374-P), Spain, and by a grant from CONACYT (CONACYT-UAG I0110/62/10), México.Publicad

La resolución de problemas: un recurso para el desarrollo de la formación de la personalidad

En el artículo se propone una caracterización del concepto problema a partir del estudio de las diferentes concepciones didácticas desarrolladas hasta la actualidad. Además, el trabajo aporta una clasificación de problemas así como la características de los mismos para incidir en el desarrollo de la personalidad en general y en los valores en particular. Además como soporte a los elementos teóricos desarrollados se plantean un conjunto de problemas tipos

New inequalities involving the geometric-arithmetic index

Let G = (V, E) be a simple connected graph and di be the degree of its ith vertex. In a recent paper [J. Math. Chem. 46 (2009) 1369-1376] the first geometricarithmetic index of a graph G was defined as GA1 = X uv∈E 2 √ dudv du + dv . This graph invariant is useful for chemical proposes. The main use of GA1 is for designing so-called quantitative structure-activity relations and quantitative structureproperty relations. In this paper we obtain new inequalities involving the geometricarithmetic index GA1 and characterize the graphs which make the inequalities tight. In particular, we improve some known results, generalize other, and we relate GA1 to other well-known topological indices.We are grateful to the constructive comments from anonymous referee on our pape. The first and third authors are supported by the "Ministerio de Economía y Competititvidad" (MTM2013-46374-P and MTM2015-69323-REDT), Spain, and by the CONACYT (FOMIX-CONACyT-UAGro 249818), Mexico

Domination on hyperbolic graphs

If k ≥ 1 and G = (V, E) is a finite connected graph, S ⊆ V is said a distance k-dominating set if every vertex v ∈ V is within distance k from some vertex of S. The distance k-domination number γ kw (G) is the minimum cardinality among all distance k-dominating sets of G. A set S ⊆ V is a total dominating set if every vertex v ∈ V satisfies δS (v) ≥ 1 and the total domination number, denoted by γt(G), is the minimum cardinality among all total dominating sets of G. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this paper we obtain relationships between the hyperbolicity constant δ(G) and some domination parameters of a graph G. The results in this work are inequalities, such as γkw(G) ≥ 2δ(G)/(2k + 1) and δ(G) ≤ γt(G)/2 + 3.Supported by two grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain, and a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain

New Hermite-Hadamard Type Inequalities Involving Non-Conformable Integral Operators

At present, inequalities have reached an outstanding theoretical and applied development and they are the methodological base of many mathematical processes. In particular, Hermite– Hadamard inequality has received considerable attention. In this paper, we prove some new results related to Hermite–Hadamard inequality via symmetric non-conformable integral operators.This paper was supported in part by two grants from the Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227- C2-1-P and MTM2017-90584-REDT), Spain