Searcharxiv⌕ Search

arXiv subjects

Jose M. Rodriguez

Publications and source records attributed to Jose M. Rodriguez.

13 recordsLinked to original sources
arXiv2025

Toponogov comparison and the collar theorem for complete surfaces with an appendix on the level sets of distance functions

In the 1970s, the collar theorem was proven, establishing the existence of uniform tubular neighborhoods of simple closed geodesics on compact surfaces, whose widths depend only on the lengths of the geodesics and the lower bound of the curvature, but not on the surface. In this paper, we improve this result by eliminating the compactness hypothesis. To achieve this result, we needed to prove new Toponogov-type triangle comparison theorems. We also add a new theorem to the literature on the rectifiabilty of the level sets of the distance function, with the corollary that on thin infinite cylinders with geodesic boundary all sets of constant distance to the boundary are simple closed Lipschitz curves.

math.DG↗
arXiv2023

Multiplicative topological indices: Analytical properties and application to random networks

We make use of multiplicative degree-based topological indices $X_Π(G)$ to perform a detailed analytical and statistical study of random networks $G=(V(G),E(G))$. We consider two classes of indices: $X_Π(G) = \prod_{u \in V(G)} F_V(d_u)$ and $X_Π(G) = \prod_{uv \in E(G)} F_E(d_u,d_v)$, where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $F_V(x)$ and $F_E(x,y)$ are functions of the vertex degrees. Specifically, we find analytical inequalities involving these multiplicative indices. Also, we apply $X_Π(G)$ on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. We show that $\left< \ln X_Π(G) \right>$, normalized to the order of the network, scale with the corresponding average degree; here $\left< \cdot \right>$ denotes the average over an ensemble of random networks.

math.CO↗
arXiv2021

Normalized Sombor indices as complexity measures of random graphs

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: Erdös-Rényi graphs, random geometric graphs, and bipartite random graphs. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the graph, scale with the graph average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random graphs and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the graph adjacency matrix.

math.CO↗
arXiv2021

Analytical and computational study of the variable inverse sum deg index

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 applicative properties. The aim of this paper is to obtain new inequalities for the variable inverse sum deg index, and to characterize graphs extremal with respect to them. Some of these inequalities generalize and improve previous results for the inverse sum deg index. In addition, we computationally validate some of the obtained inequalities on ensembles of random graphs and show that the ratio $\left\langle ISD_a(G) \right\rangle/n$ ($n$ being the order of the graph) depends only on the average degree $\left\langle d \right\rangle$.

math.CO↗
arXiv2021

Analytical and computational properties of the variable symmetric division deg index

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $SDD_α(G) = \sum_{uv \in E(G)} (d_u^α/d_v^α+d_v^α/d_u^α)$, 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 vertex $u$, and $α\in \mathbb{R}$. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $SDD_α(G)$ index on random graphs and show that the ratio $\left\langle SDD_α(G) \right\rangle/n$ ($n$ being the order of the graph) depends only on the average degree $\left\langle d \right\rangle$.

math.CO↗
arXiv2020

Several extremal problems on graphs involving the circumference, girth, and hyperbolicity constant

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 $c$, and $m$ edges. In this work, we study the four following extremal problems on graphs: $A(g,c,n)=\min\{δ(G)\,|\; G \in \mathcal{G}(g,c,n) \}$, $B(g,c,n)=\max\{δ(G)\,|\; G \in \mathcal{G}(g,c,n) \}$, $α(g,c,m)=\min\{δ(G)\,|\; \in \mathcal{H}(g,c,m) \}$ and $β(g,c,m)=\max\{δ(G)\,|\; G \in \mathcal{H}(g,c,m) \}$. In particular, we obtain bounds for $A(g,c,n)$ and $α(g,c,m)$, and we compute the precise value of $B(g,c,n)$ and $β(g,c,m)$ for all values of $g$, $c$, $n$ and $m$.

math.CO↗
arXiv2020

A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs

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 particular, we prove that having a pole is a necessary condition and, therefore, it can be removed as hypothesis.

math.DG↗
arXiv2020

Computational and analytical studies of the Randić index in Erdös-Rényi models

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 $p \in (0,1)$. First, from a detailed scaling analysis, we show that $\left\langle \overline{R}(G) \right\rangle = \left\langle R(G)\right\rangle/(n/2)$ scales with the product $ξ\approx np$, so we can define three regimes: a regime of mostly isolated vertices when $ξ< 0.01$ ($R(G)\approx 0$), a transition regime for $0.01 < ξ< 10$ (where $0 10$ ($R(G)\approx n/2$). Then, motivated by the scaling of $\left\langle \overline{R}(G) \right\rangle$, we analytically (i) obtain new relations connecting $R(G)$ with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on $R(G)$ for graphs in Erdös-Rényi models.

cond-mat.dis-nn↗
arXiv2015

Bounds on Gromov 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 $[x_{1}x_{2}]$, $[x_{2}x_{3}]$ and $[x_{3}x_{1}]$ 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 $X$. If $X$ is hyperbolic, we denote by $δ(X)$ the sharp hyperbolicity constant of $X$, i.e. $δ(X) =\inf \{ δ\geq 0:{0.3cm}$ X ${0.2cm}$ $\text{is} {0.2cm} δ\text{-hyperbolic} \}.$ To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolycity constant in terms of some parameters of the graph. Denote by $\mathcal{G}(n,m)$ the set of graphs $G$ with $n$ vertices and $m$ edges, and such that every edge has length $1$. In this work we estimate $A(n,m):=\min\{δ(G)\mid G \in \mathcal{G}(n,m) \}$ and $B(n,m):=\max\{δ(G)\mid G \in \mathcal{G}(n,m) \}$. In particular, we obtain good bounds for $B(n,m)$, and we compute the precise value of $A(n,m)$ for all values of $n$ and $m$. Besides, we apply these results to random graphs.

math.CO↗
arXiv2009

Computation of conformal representations of compact Riemann surfaces

We find a system of two polynomial equations in two unknowns, whose solution allows to give an explicit expression of the conformal representation of a simply connected three sheeted compact Riemann surface onto the extended complex plane. This function appears in the description of the ratio asymptotic of multiple orthogonal polynomials with respect to so called Nikishin systems of two measures.

math.CV↗
arXiv2008

A real variable characterization of Gromov hyperbolicity of flute surfaces

In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This theorem has several consequences; in particular, it allows to deduce a result about stability of hyperbolicity, even though the original surface and the modified one are not quasi-isometric.

math.CV↗
arXiv2008

Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics

We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows to deduce the non-hyperbolicity of any periodic Denjoy domain.

math.CV↗