# Jose M RodriguezUniversity Carlos III de Madrid | UC3M · High Technical College

Jose M Rodriguez

PhD

## About

228

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## Publications

Publications (228)

Inequalities play a main role in pure and applied mathematics. In this paper, we prove a generalization of Milne inequality for any measure space. The argument in the proof of this inequality allows us to obtain other Milne-type inequalities. Also, we improve the discrete version of Milne inequality, which holds for any positive value of the parame...

Herein we study p-parabolicity on graphs. We prove that if a uniform graph satisfies the (Cheeger) isoperimetric inequality, then it is non-p-parabolic for every \(1<p<\infty \). Moreover, we give sufficient conditions for a uniform graph to be non-parabolic and to be p-parabolic for every \(1<p<\infty \).

The harmonic polynomial was defined in order to understand better the harmonic topological index. Here, we obtain several properties of this polynomial, and we prove that several properties of a graph can be deduced from its harmonic polynomial. Also, we prove that graphs with the same harmonic polynomial share many properties although they are not...

Inequalities are essential in pure and applied mathematics. In particular, Opial’s inequality and its generalizations have been playing an important role in the study of the existence and uniqueness of initial and boundary value problems. In this work, some new Opial-type inequalities are given and applied to generalized Riemann-Liouville-type inte...

The concept of Gromov hyperbolicity is a geometric concept that leads to a rich general theory. Johnson and Kneser graphs are interesting combinatorial graphs defined from systems of sets. In this work we compute the precise value of the hyperbolicity constant of every Johnson graph. Also, we obtain good bounds on the hyperbolicity constant of ever...

Objective: The re-emergence of psychedelics in the field of psychiatric treatment, particularly for depression, has increasingly gained the attention of the scientific community due to the large number of people suffering from depression and relatively low treatment effectiveness. This study conducted a systematic review and meta-analysis of articl...

The associations between Vitamin D deficiency and the severity of motor symptoms, coupled with positive outcomes from supplementation, highlight the clinical relevance of Vitamin D in managing Amyotrophic Lateral Sclerosis (ALS), Parkinson's Disease (PD), and Frontotemporal Dementia (FTD). Further research is warranted to unravel the specific mecha...

Large scale properties of Riemannian manifolds, in particular, those properties preserved by quasi-isometries, can be studied using discrete structures which approximate the manifolds. In a sequence of papers, M. Kanai proved that, under mild conditions, many properties are preserved by a certain (quasi-isometric) graph approximation of a manifold....

We consider two general classes of multiplicative degree-based topological indices (MTIs), denoted by $ X_{\Pi, F_V}(G) = \prod_{u \in V(G)} F_V(d_u) $ and $ X_{\Pi, F_E}(G) = \prod_{uv \in E(G)} F_E(d_u, d_v) $, where $ uv $ indicates the edge of $ G $ connecting the vertices $ u $ and $ v $, $ d_u $ is the degree of the vertex $ u $, and $ F_V(x)...

The inverse degree index, also called inverse index, first attracted attention through numerous conjectures generated by the computer programme Graffiti. Since then its relationship with other graph invariants has been studied by several authors. In this paper we obtain new inequalities involving the inverse degree index, and we characterize graphs...

In this paper, we establish several HHölder -type inequalities using Jensen-type and Young-type inequalities as key tools. Particularly noteworthy is a reverse Hölder inequality with
the Specht’s ratio. Furthermore, we obtain a reverse Young-type inequality and we apply these results to the fractional context, both globally and locally.

Topological indices are used to understand physicochemical properties of chemical compounds,since they capture some properties of a molecule in a single number. The \emph{sum lordeg index} is defined as$$SL(G) = \sum_{u\in V(G)} d_u \sqrt{\log d_u} \,.$$This index is interesting from an applied viewpoint sinceit is the best predictor of octanol-wat...

The inverse degree index, also called inverse index, first attracted attention through numerous conjectures generated by the computer programme Graffiti.Since then its relationship with other graph invariants has been studied by several authors. In this paper we obtain new inequalities involving the inverse degree index, and we characterize graphs...

In this paper, we introduce the theory of a generalized Fourier transform in order to solve differential equations with a generalized fractional derivative, and we state its main properties. In particular, we obtain the new corresponding convolution, inverse and Plancherel formulas, and Hausdorff–Young type inequality. We show that this generalized...

We make use of multiplicative degree-based topological indices $X_\Pi(G)$ to perform a detailed analytical and statistical study of random networks $G=(V(G),E(G))$. We consider two classes of indices: $X_\Pi(G) = \prod_{u \in V(G)} F_V(d_u)$ and $X_\Pi(G) = \prod_{uv \in E(G)} F_E(d_u,d_v)$, where $uv$ denotes the edge of $G$ connecting the vertice...

Let G=(V(G),E(G)) be a simple graph and denote by du the degree of the vertex u∈V(G). Using a geometric approach, I. Gutman introduced a new vertex-degree-based topological index, defined asSO(G)=∑uv∈E(G)(du)2+(dv)2,and named Sombor index. It is a molecular descriptor with an impressive research activity in recent years. In this paper we propose an...

In this paper, we show a complete characterization of the uniform boundedness of the partial sum operator in a discrete Sobolev space with Jacobi measure. As a consequence, we obtain the convergence of the Fourier series. Moreover it is showed that this Sobolev space is the first category which implies that it is not possible to apply the Banach–St...

In this paper, the authors work with a general formulation of the fractional derivative of Caputo type. They study oscillatory solutions of differential equations involving these general fractional derivatives. In particular, they extend the Kamenev-type oscillation criterion given by Baleanu et al. in 2015. In addition, we prove results on the exi...

The inverse degree index, also called inverse index, first attracted attention through numerous conjectures generated by the computer programme Graffiti. Since then its relationship with other graph invariants has been studied by several authors. In this paper we obtain new inequalities involving the inverse degree index, and we characterize graphs...

Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including Liouville property) between Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai's hypotheses in many cases are not satisfied in the co...

The complementary prism of G$$ G $$, denoted by GG‾$$ G\overline{G} $$, is the graph obtained from the disjoint union of G$$ G $$ and G‾$$ \overline{G} $$ by adding edges between the corresponding vertices of G$$ G $$ and G‾$$ \overline{G} $$. In this paper, we study the hyperbolicity constant of GG‾$$ G\overline{G} $$. In particular, we obtain upp...

Some years ago, the harmonic polynomial was introduced in order to understand better the harmonic topological index; for instance, it allows to obtain bounds of the harmonic index of the main products of graphs. Here, we obtain several properties of this polynomial, and we prove that several properties of graphs can be deduced from their harmonic p...

In this work we obtain inequalities relating a general topological index of a graph, $F(G)= \sum_{uv\in E(G)} f(d_u,d_v)$, with its corresponding exponential index, $e^F(G)= \sum_{uv\in E(G)} e^{f(d_u,d_v)}$ when a (symmetric) function $f(\cdot,\cdot)$ is evaluated in the degree of the two adjacent vertices to each edge. Besides, we relate two gene...

Inequalities play a major role in pure and applied mathematics. In particular, the inequality plays an important role in the study of Rosseland’s integral for the stellar absorption. In this paper we obtain new Milne-type inequalities, and we apply them to the generalized Riemann–Liouville-type integral operators, which include most of the known Ri...

For a geodesic metric space X and for 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 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. If X is hyperbolic, we...

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_1 x_2] $, $ [x_2 x_3] $ and $ [x_3 x_1] $ in $ X $. The space $ X $ is hyperbolic if there exists a constant $ \delta \ge 0 $ such that any side of any geodesic triangle in $ X $ is contained in the...

In this paper, we perform analytical and statistical studies of Revan indices on graphs G: R(G) = uv∈E(G) F(r u , r v), where uv denotes the edge of G connecting the vertices u and v, r u is the Revan degree of the vertex u, and F is a function of the Revan vertex degrees. Here, r u = ∆ + δ − d u with ∆ and δ the maximum and minimum degrees among t...

Inequalities play an important role in pure and applied mathematics. In particular, Jensen’s inequality, one of the most famous inequalities, plays the 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 functi...

In this paper we prove several optimal inequalities involving the hyperbolicity constant of complementary prisms networks. Moreover, we obtain bounds and closed formulas for the general topological indices A(G)=∑ a(d,d) and B(G)=∑b(d) of complementary prisms networks.

It is well known that mathematical inequalities have played a very important role in solving both theoretical and practical problems. In this paper, we show some new results related to Ostrowski type inequalities for generalized integral operators.

In this paper, we introduce the theory of a generalized Fourier transform in order to solve differential equations with a generalized fractional derivative, and we state its main properties. In particular, we obtain the corresponding convolution, inverse and Plancherel formulas, and Hausdorff-Young inequality. We show that this generalized Fourier...

We obtain inequalities relating a general topological index of a graph, $F(G)=\sum_{u v \in E(G)} f\left(d_u, d_v\right)$, with its corresponding exponential index, $e^F(G)=\sum_{u v \in E(G)} e^{f\left(d_u, d_v\right)}$ when a (symmetric) function $f(\cdot, \cdot)$ is evaluated in the degree of the two adjacent vertices to each edge. Besides, we r...

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $ SDD_\alpha(G) = \sum_{uv \in E(G)} (d_u^\alpha/d_v^\alpha+d_v^\alpha/d_u^\alpha) $, and to characterize graphs extremal with respect to them. Here, by $ uv $ we mean the edge of a graph $ G $ joining the vertices $ u $ and $ v $, and $ d_u $ denotes t...

In this paper, we present a definition of the generalized fractional local derivative, which contains as a particular case several of those reported in the literature. In this framework, we apply this operator to the study of the solutions of some fractional differential equations and we focus mainly on a Gompertz model applied to the study of tube...

Inequalities play an important role in pure and applied mathematics. In particular, Opial inequality plays a main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. It has several interesting generalizations. In this work we prove some new Opial-type inequalities, and we apply them t...

Kanai proved the stability under quasi‐isometries of numerous global properties (including existence of Green's function, i.e., non‐parabolicity) between Riemannian manifolds of bounded geometry. Unfortunately, Kanai's hypotheses are not usually satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we prove the...

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.

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...

In this paper we study lower bounds in a unified way for a large family of topological indices, including the first variable Zagreb index M1α. Our aim is to obtain sharp inequalities and characterize the corresponding extremal graphs. The main results provide lower bounds for several vertex-degree-based topological indices. These bounds are new eve...

In this paper, we present a general formulation of the well-known fractional drifts of Riemann-Liouville type. We state the main properties of these integral operators. Besides, we study Ostrowski, Székely-Clark-Entringer and Hermite-Hadamard-Fejér inequalities involving these general fractional operators.

We introduce a definition of a generalized conformable derivative of order α > 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...

We introduce a degree-based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: $mSO_\alpha(G) = \sum_{uv \in E(G)} \left[\left( d_u^\alpha+d_v^\alpha \right) /2 \right]^{1/\alpha}$. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the deg...

It is known that the problem of computing the adjacency dimension of a graph is NP-hard. This suggests finding the adjacency dimension for special classes of graphs or obtaining good bounds on this invariant. In this work we obtain general bounds on the adjacency dimension of a graph G in terms of known parameters of G. We discuss the tightness of...

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 work we obtain new lower and upper optimal bounds of general Sombor indices. Specifically, we have 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.

The Sombor index SO, the reduced Sombor index SOred, and the average Sombor index SOavg have been introduced recently, and it was shown that they have good predictive potential in Mathematical Chemistry. We obtain in this work several new lower and upper bounds of these indices. Specifically, we find inequalities between SO and SOred and characteri...

The study of Gromov hyperbolic graphs has many applications. In this paper we study the hyperbolicity constant of hexagonal systems. In particular, we compute the hyperbolicity constant of every catacondensed hexagonal system. Besides, we obtain upper and lower bounds of general hexagonal systems. Since the hyperbolicity constant of a graph measure...

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor...

We study in this paper the relationship of isoperimetric inequality and hyperbolicity for graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian manifolds (with bounded local geometry) satisfying the (Cheeger) isoperimetric inequality, in terms of their Gromov boundary, improving similar results from a previous work....

We perform a detailed computational study of the recently introduced Sombor indices on random graphs. Specifically, we apply Sombor indices on three models of random graphs: Erd\"os-R\'enyi graphs, random geometric graphs, and bipartite random graphs. Within a statistical random matrix theory approach, we show that the average values of Sombor indi...

A large number of graph invariants of the form $\sum_{uv \in E(G)} F(d_u,d_v)$ are studied in mathematical chemistry, where $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, and $d_u$ is the degree of the vertex $u$. Among them the variable inverse sum deg index $ISD_a$, with $F(d_u,d_v)=1/(d_u^a+d_v^a)$, was found to have...

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $SDD_\alpha(G) = \sum_{uv \in E(G)} (d_u^\alpha/d_v^\alpha+d_v^\alpha/d_u^\alpha)$, and to characterize graphs extremal with respect to them. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the ve...

In this work, we obtained new results relating the generalized atom-bond connectivity index with the general Randić index. Some of these inequalities for ABC α improved, when α = 1/2, known results on the ABC index. Moreover, in order to obtain our results, we proved a kind of converse Hölder inequality, which is interesting on its own.

Recently, the arithmetic–geometric index (AG) was introduced, inspired by the well-known and studied geometric–arithmetic index (GA). In this work, we obtain new bounds on the arithmetic–geometric index, improving upon some already known bounds. In particular, we show families of graphs where such bounds are attained.

In this work we perform analytical and statistical studies of the Rodríguez–Velázquez (RV) indices on graphs G. The topological RV(G) indices, recently introduced in Rodríguez–Velázquez and Balaban (J Math Chem 57:1053, 2019), are based on graph adjacency matrix eigenvalues and eigenvectors. First, we analytically obtain new relations connecting RV...

The concept of arithmetic-geometric index was recently introduced in chemical graph theory, but it has proven to be useful from both a theoretical and practical point of view. The aim of this paper is to obtain new bounds of the arithmetic-geometric index and characterize the extremal graphs with respect to them. Several bounds are based on other i...

In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. A...

It is known that complete Riemannian surfaces can be obtained by pasting three kinds of pieces. In this paper we prove an analogous result in the context of plane domains with their quasihyperbolic metrics. In order to do it, we prove several facts about quasihyperbolic closed geodesics of independent interest; for instance, we characterize the exi...

The aim of this paper is to obtain new inequalities for a large family of topological indices restricted to unicyclic graphs and to characterize the set of extremal unicyclic graphs with respect to them. This family includes variable first Zagreb, variable sum exdeg, multiplicative second Zagreb and Narumi-Katayama indices. Our main results provide...

Kanai proved that quasi-isometries between Riemannian manifolds with bounded geometry preserve many global properties, including the existence of Green’s function, i.e., non-parabolicity. However, Kanai’s hypotheses are too restrictive. Herein we prove the stability of p-parabolicity (with \(1<p<\infty \)) by quasi-isometries between Riemannian man...

The atom-bond connectivity and the generalized atom-bond connectivity indices have shown to be useful in the QSPR/QSAR researches. The aim of this paper is to obtain new inequalities for these indices and characterize graphs extremal with respect to them.

We obtain inequalities involving many topological indices in classical graph products by using the f-polynomial. In particular, we work with lexicographic product, Cartesian sum and Cartesian product, and with first Zagreb, forgotten, inverse degree and sum lordeg indices.

The atom-bond connectivity index ABC is a topological index that has recently found remarkable applications in the study of the strain energy of cycloalkanes as well as in rationalizing the stability of linear and branched alkanes. In order to improve the correlation properties of the atom-bond connectivity index for the heat of formation of alkane...

The concept of arithmetic-geometric index was introduced in the chemical graph theory recently, but it has proven to be useful from both a theoretical and practical point of view. The aim of this paper is to obtain new bounds of the arithmetic-geometric index and characterize graphs extremal with respect to them.

Topological indices are useful for predicting the physicochemical behavior of chemical compounds. A main problem in this topic is finding good bounds for the indices, usually when some parameters of the graph are known. The aim of this paper is to use a unified approach in order to obtain several new inequalities for a wide family of topological in...

In this work we perform computational and analytical studies of the Randić index R(G) in Erdös–Rényi models G(n, p) characterized by n vertices connected independently with probability p ∈ (0, 1). First, from a detailed scaling analysis, we show that 〈R¯(G)〉=〈R(G)〉/(n/2) scales with the product ξ ≈ np, so we can define three regimes: a regime of mo...

In this paper, we use a conformable fractional derivative G𝛼T, with kernel
T(t, 𝛼) = exp((𝛼−1)t), in order to study the fractional differential equation associated to a logistic growthmodel.As a practical application,we estimate the order of the derivative of the fractional logisticmodels, by solving an inverse problem involving real data. In the s...

In this paper, we use the generalized fractional derivative in order to study the fractional differential equation associated with a fractional Gaussian model. Moreover, we propose new properties of generalized differential and integral operators. As a practical application, we estimate the order of the derivative of the fractional Gaussian models...

A celebrated theorem of Kanai states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds (with positive injectivity radius) and graphs. Our main result states that we can study the (Cheeger) isoperimetric inequality in a Riemann surface by using a graph related to it, even if the surface has injectivity ra...

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 k w (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 δ S (v...

To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let $\mathcal{G}(g,c,n)$ be the set of graphs $G$ with girth $g(G)=g$, circumference $c(G)=c$, and $n$ vertices; and let $\mathcal{H}(g,c,m)$ be the set of graphs with girth $g$, circumference $c$,...

Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. It measures the tree-likeness of a graph from a metric viewpoint. In particular, we are interested in circular-arc graphs, which is an important class of geometric intersection graphs. In this paper we give sharp bounds fo...

In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary improving similar results from a previous work...

In this work we perform computational and analytical studies of the Randi\'c index $R(G)$ in Erd\"os-R\'{e}nyi models $G(n,p)$ characterized by $n$ vertices connected independently with probability $p \in (0,1)$. First, from a detailed scaling analysis, we show that $\left\langle \overline{R}(G) \right\rangle = \left\langle R(G)\right\rangle/(n/2)$...

In this paper we use a conformable fractional derivative $G_{T}^{\alpha }$, with kernel $T(t,\alpha)={ e }^{(\alpha-1)t}$, in order to study the fractional differential equation associated to a logistic growth model. As a practical application, we estimate the order of the derivative of the fractional logistic models, by solving an inverse problem...

In this paper, we generalize the classical definition of Gromov hyperbolicity to the context of directed graphs and we extend one of the main results of the theory: the equivalence of the Gromov hyperbolicity and the geodesic stability. This theorem has potential applications to the development of solutions for secure data transfer on the internet.

Using the symmetry property of the inverse degree index, in this paper, we obtain several mathematical relations of the inverse degree polynomial, and we show that some properties of graphs, such as the cardinality of the set of vertices and edges, or the cyclomatic number, can be deduced from their inverse degree polynomials.

Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. In particular, analytic properties of such polynomials have been extensively studied, mainly focused on their asymptotic behavior and the location of their zeros. On the other hand, the behavior of the Fourier–Sobolev projector allows to deal with very interest...

Given any function f : Z + → R + , let us define the f-index I f ( G ) = ∑ u ∈ V ( G ) f ( d u ) and the f-polynomial P f ( G , x ) = ∑ u ∈ V ( G ) x 1 / f ( d u ) − 1 , for x > 0 . In addition, we define P f ( G , 0 ) = lim x → 0 + P f ( G , x ) . We use the f-polynomial of a large family of topological indices in order to study mathematical relat...

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 i...

In this paper we use a conformable fractional derivative $G_{T}^{\alpha }$, through the kernel $T(t,\alpha)={ e }^{(\alpha-1)t}$; in order to study a logistic growth model. In addition, we study the fractional differential equation associated to the logistic model. As a practical application, we estimate the order of the derivative of the fractiona...

In this paper, newly proposed generalized fractional derivative has been used to study the classical Drude model. This model describe the motion of the electrons in a metal in the presence of an external electric field. We present the novel fractional operator and the general conditions for the existence and the uniqueness of the exact solutions fo...

The aim of this paper is to obtain new inequalities involving some topological indices of a graph and characterize graphs extremal with respect to them. Our main results provide lower bounds on several indices involving just the minimum and the maximum degree of the graph G. This family of indices includes, among others, the Wiener index and severa...

The inverse degree index, also called inverse index, first attracted attention through numerous conjectures generated by the computer programme Graffiti. Since then, its relationship with other graph invariants has been studied by several authors. In this paper, we obtain new inequalities involving the inverse degree index, and we characterize grap...

Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including the volume growth rate) between non-bordered Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai’s hypotheses are not usually satisfie...

In this paper we give a lower bound for the visual Hausdorff dimension of the geodesics escaping through different ends of Riemannian surfaces with pinched negative curvature. This allows to show that in any Riemannian surface with pinched negative curvature and infinite area there is a large set of geodesics escaping to infinity.