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Inequalities play an important role in pure and applied mathematics. In particular, Jensen's inequality, one of the most famous inequalities, plays a main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. In this work we prove some new Jensen-type inequalities for m-convex functions, and apply them to generalized Riemann-Liouville-type integral operators. Furthermore, as a remarkable consequence, some new inequalities for convex functions are obtained. The research of Yamilet Quintana, José M. Rodríguez, and José M. Sigarreta is supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/10.13039/501100011033), Spain. The research of Yamilet Quintana and José M. Rodríguez is supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23) and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).

In this work we obtain new lower and upper optimal bounds of general Sombor indices. Specifically, we get inequalities for these indices relating them with other indices: the first Zagreb index, the forgotten index and the first variable Zagreb index. Finally, we solve some extremal problems for general Sombor indices. We would like to thank the reviewers by their careful reading of the manuscript and their suggestions which have improved the presentation of this work. The research of José M. Rodríguez and José M. Sigarreta was supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/10.13039/501100011033), Spain. The research of Jose M. Rodríguez is supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).

A main topic in the study of topological indices is to find bounds of the indices involving several parameters and/or other indices. In this paper we perform statistical (numerical) and analytical studies of the harmonic index H(G), and other topological indices of interest, on Erdos-Rényi (ER) graphs G(n, p) characterized by n vertices connected independently with probability p ∈ (0, 1). Particularly, in addition to H(G), we study here the (−2) sum-connectivity index χ−2(G), the modified Zagreb index MZ(G), the inverse degree index ID(G) and the Randic index R(G). First, to perform the statistical study of these indices, we define the averages of the normalized indices to their maximum value: {H(G)}, {χ−2(G)}, {MZ(G)}, {ID(G)}, {R(G)}. Then, from a detailed scaling analysis, we show that the averages of the normalized indices scale with the product ξ ≈ np. Moreover, we find two different behaviors. On the one hand, hH(G)i and hR(G)i, as a function of the probability p, show a smooth transition from zero to n/2 as p increases from zero to one. Indeed, after scaling, it is possible to define three regimes: a regime of mostly isolated vertices when ξ < 0.01 (H(G), R(G) ≈ 0), a transition regime for 0.01 < ξ < 10 (where 0 < H(G), R(G) < n/2), and a regime of almost complete graphs for ξ > 10 (H(G), R(G) ≈ n/2). On the other hand, hχ−2(G)i, hMZ(G)i and hID(G)i increase with p until approaching their maximum value, then they decrease by further increasing p. Thus, after scaling the curves corresponding to these indices display bell-like shapes in log scale, which are symmetric around ξ ≈ 1; i.e. the percolation transition point of ER graphs. Therefore, motivated by the scaling analysis, we analytically (i) obtain new relations connecting the topological indices H, χ−2, MZ, ID and R that characterize graphs which are extremal with respect to the obtained relations and (ii) apply these results in order to obtain inequalities on H, χ−2, MZ, ID and R for graphs in ER models. J.A.M.-B. acknowledges financial support from FAPESP (Grant No. 2019/ 06931-2), Brazil, CONACyT (Grant No. 2019-000009-01EXTV-00067) and PRODEP-SEP (Grant No. 511-6/2019.-11821), Mexico. J.M.R. and J.M.S. acknowledge financial support from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/ 10.13039/501100011033), Spain.

We introduce a definition of a generalized conformable derivative of order alfa > 0 (where this parameter does not need to be integer), with which we overcome some deficiencies of known local derivatives, conformable or not. This definition allows us to compute fractional derivatives of functions defined on any open set on the real line (and not just on the positive half- line). Moreover, we extend some classical results to the context of fractional derivatives. Also, we obtain results for the case alfa > 1 The research of José M. Rodríguez and José M. Sigarreta is supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI /10.13039/501100011033), Spain.

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.

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.

El modelo didáctico propuesto favorece el proceso de formación de los valores en los estudiantes, a través de la resolución de problemas matemáticos. Las etapas del modelo permiten constatar el estado de los estudiantes en relación con los conocimientos que poseen en torno a los valores y su multiplicidad. Además, estimula el proceso de objetivación axiológica a partir del control y la evaluación constante de la situación valoral en el universo estudiantil al que se dirige el proceso formativo, adjunto al logro de las precisiones que evidencia la esencia de los valores; así hay una visión objetiva de los mismos, facilita la labor de los docentes en cuanto a su tratamiento y contribuye a desentrañar y desarrollar sus perspectivas axiológicas.

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. Publicado

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. Publicado

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.

The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space and Riemannian manifolds of negative sectional curvature. It is remarkable that a simple concept leads to such a rich general theory. 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 Ph. D. Thesis we characterize the hyperbolicity constant of interval graphs and circular-arc graphs. Likewise, we provide relationships between dominant sets and the hyperbolicity constant. Finally, we study the invariance of the hyperbolicity constant when the graphs are transformed by several operators. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Lojo

Topological indices have been widely used in different fields associated with scientific research. They are recognized as useful tools in applied research in Chemistry, Ecology, Biology, Physics, among others. For many years, scientists have been trying to improve the predictive power of the famous Randi’c index. This led to the introduction and study of new topological descriptors that correlate or improve the level of prediction of the Randi’c index. Among the most commonly used descriptors are the Inverse index, the first general Zagreb index and the recently introduced Arithmetic- Geometric index. In this work we study the mathematical properties and relationships of the aforementioned topological indices. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Lojo

In this work, we introduce a definition of a local fractional derivative and a fractional integral of order alfa > 0 (where this parameter does not need to be integer), with which we overcome some deficiencies of known local derivatives, conformable or not. This definition allows to compute fractional derivatives of functions defined on any open set on the real line (and not just on the positive half-line). Moreover, we extend some classical results to the context of fractional derivatives. Also, applications of the fractional derivative through the direct and inverse problems are shown, as well as the feasibility of the fractional calculus in real and simulated problems. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Domingo de Guzmán Pestana Galván.- Secretario: Ana Portilla Ferreira.- Vocal: Ana Granados Sanandrés

A lo largo de esta tesis trabajaremos con gráficas hiperbólicas. La teoría de gráficas es una línea de investigación de las Matemáticas Discretas. A pesar de ser un área relativamente reciente, está creciendo muy rápidamente con muchos resultados descubiertos en las ultimas 3 décadas. Dicha teoría permite tratar en forma más simple, cualquier problema donde exista una relación binaria entre ciertos objetos, por lo que sus usos son bastantes amplios. Muestra de ello es que podemos encontrar aplicaciones a áreas dentro de las mismas Matemáticas, Ingeniería, Biología, Sociología, Administración, etc.

Hyperbolic spaces, defined by Gromov in, play an important role in geometric group theory and in the geometry of negatively curved spaces. The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, Riemannian manifolds of negative sectional curvature bounded away from 0, and of discrete spaces like trees and the Cayley graphs of many finitely generated groups. It is remarkable that a simple concept leads to such a rich general theory. The first works on Gromov hyperbolic spaces deal with finitely generated groups. Initially, Gromov spaces were applied to the study of automatic groups in the science of computation; indeed, hyperbolic groups are strongly geodesically automatic, i.e., there is an automatic structure on the group. The concept of hyperbolicity appears also in discrete mathematics, algorithms and networking. For example, it has been shown empirically in that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension; furthermore, it is evidenced that many real networks are hyperbolic. A few algorithmic problems in hyperbolic spaces and hyperbolic graphs have been considered in recent papers. Another important application of these spaces is the study of the spread of viruses through the internet. Furthermore, hyperbolic spaces are useful in secure transmission of information on the network. The hyperbolicity has also been used extensively in the context of random graphs. For example, it was shown that several types of small-world networks and networks with given expected degrees are not hyperbolic in some sense. The study of Gromov hyperbolic graphs is a subject of increasing interest in graph theory; and the references therein. In our study on the hyperbolicity in graphs we use the notations (...) Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Domingo de Guzmán Pestana Galván.- Secretario: Ana María Portilla Ferreira.- Vocal: Eva Touris Lojo