arXiv subjects

José M. Sigarreta

Publications and source records attributed to José M. Sigarreta.

14 recordsLinked to original sources

arXiv2026

The quarter-point theorem for Gromov hyperbolic graphs

Gromov hyperbolicity measures the geometric complexity of metric spaces and metric graphs through geodesic triangles and four-point conditions. While calculating the exact continuous hyperbolicity constant $δ^*(G)$ of a graph $G$ typically requires evaluating quadruples of points over an infinite metric continuum, discretizing this domain is a fundamental challenge in metric graph theory. In this paper, we study the continuous Gromov hyperbolicity constant via the Gromov product. Our main result, the Quarter-Point Theorem, establishes that the search space for quadruples $(x,y,z,w)$ attaining the exact hyperbolicity constant $δ^*(G)$ can be reduced from the entire metric graph to a finite or discrete subset $J'(G)$, comprising solely the vertices, edge midpoints, and quarter-points of $G$. This result holds for every graph, finite or infinite. This theoretical result provides a considerably simplified combinatorial framework for calculating hyperbolicity in complex networks and metric graphs. Furthermore, we exploit this discretization to significantly improve classical results by establishing and structural bounds relating the global hyperbolicity constant $δ^*(G)$ with its restrictions to vertices ($δ_V^*(G)$) and midpoints ($δ_J^*(G)$).

math.CO↗

arXiv2023

Topological and spectral properties of random digraphs

We investigate some topological and spectral properties of Erdős-Rényi (ER) random digraphs $D(n,p)$. In terms of topological properties, our primary focus lies in analyzing the number of non-isolated vertices $V_x(D)$ as well as two vertex-degree-based topological indices: the Randić index $R(D)$ and sum-connectivity index $χ(D)$. First, by performing a scaling analysis we show that the average degree $\langle k \rangle$ serves as scaling parameter for the average values of $V_x(D)$, $R(D)$ and $χ(D)$. Then, we also state expressions relating the number of arcs, spectral radius, and closed walks of length 2 to $(n,p)$, the parameters of ER random digraphs. Concerning spectral properties, we compute six different graph energies on $D(n,p)$. We start by validating $\langle k \rangle$ as the scaling parameter of the graph energies. Additionally, we reformulate a set of bounds previously reported in the literature for these energies as a function $(n,p)$. Finally, we phenomenologically state relations between energies that allow us to extend previously known bounds.

cond-mat.dis-nn↗

arXiv2023

The differential on Graph Operator R(G)

Let $G=(V(G), E(G))$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. Let $S$ be a subset of $V(G)$, and let $B(S)$ be the set of neighbours of $S$ in $V(G) \setminus S$. The differential $\partial(S)$ of $S$ is the number $|B(S)|-|S|$. The maximum value of $\partial(S)$ taken over all subsets $S\subseteq V(G)$ is the differential $\partial(G)$ of $G$. The graph $R{G}$ is defined as the graph obtained from $G$ by adding a new vertex $v_e$ for each $e\in E(G)$, and by joining $v_e$ to the end vertices of $e$. In this paper we study the relationship between $\partial(G)$ and $\partial(R(G))$, and give tight asymptotic bounds for $\partial(R(G))$. We also exhibit some relationships between certain vertex sets of $G$ and $R(G)$ which involve well known graph theoretical parameters.

math.CO↗

arXiv2023

Generalized Bernoulli Differential Equation

In this paper we propose and solve a generalization of the Bernoulli Differential Equation, by means of a generalized fractional derivative. First we prove a generalization of Gronwall's inequality, which is useful for studying the stability of systems of fractional differential equations and we state results about the qualitative behavior of the trivial solution of the proposed equation. After that, we prove and state the main results about the solution of the generalized Bernoulli Differential Equation and also we give some examples that show the advantage of considering this fractional derivative approach. We also present a finite difference method as an alternative to the solution of the generalized Bernoulli equation and prove its validity by means of examples.

math.GM↗

arXiv2023

Leaking from the phase space of the Riemann-Liouville fractional standard map

In this work we characterize the escape of orbits from the phase space of the Riemann-Liouville (RL) fractional standard map (fSM). The RL-fSM, given in action-angle variables, is derived from the equation of motion of the kicked rotor when the second order derivative is substituted by a RL derivative of fractional order $α$. Thus, the RL-fSM is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the RL derivative, respectively. Indeed, for $α=2$ and given initial conditions, the RL-fSM reproduces Chirikov's standard map. By computing the survival probability $P_{\text{S}}(n)$ and the frequency of escape $P_{\text{E}}(n)$, for a hole of hight $h$ placed in the action axis, we observe two scenarios: When the phase space is ergodic, both scattering functions are scale invariant with the typical escape time $n_{\text{typ}}=\exp\langle \ln n \rangle \propto (h/K)^2$. In contrast, when the phase space is not ergodic, the scattering functions show a clear non-universal and parameter-dependent behavior.

nlin.CD↗

arXiv2022

On Opial-type inequalities via fractional calculus

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 to generalized Riemann-Liouville-type integral operators.

math.CA↗

arXiv2020

Distinguishing wheel graphs by the alliance polynomial

Distinctive power of the alliance polynomial has been studied in previous works, for instance, it has been proved that the empty, path, cycle, complete, complete without one edge and star graphs are characterized by its alliance polynomial. Moreover, it has been proved that the family of alliance polynomial of regular graphs with small degree is a very special one, since it does not contain alliance polynomial of graphs other than regular graph with the same degree. In this work we prove that the alliance polynomial also determines the wheel graphs.

math.CO↗

arXiv2017

A new approximation to 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 $GA_1$ and characterize graphs extremal with respect to them.

math.CO↗

arXiv2015

Alliance polynomial of regular graphs

The alliance polynomial of a graph $G$ with order $n$ and maximum degree $Δ$ is the polynomial $A(G; x) = \sum_{k=-Δ}^Δ A_{k}(G) \, x^{n+k}$, where $A_{k}(G)$ is the number of exact defensive $k$-alliances in $G$. We obtain some properties of $A(G; x)$ and its coefficients for regular graphs. In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of $Δ$-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not $Δ$-regular. By using this last result and direct computation we find that the alliance polynomial determines uniquely each cubic graph of order less than or equal to $10$.

math.CO↗

arXiv2015

Gromov hyperbolicity of minor graphs

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 $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ 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 $X$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In the context of graphs, to remove and to contract an edge of a graph are natural transformations. The main aim in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph $G \setminus e$ (respectively, $\,G/e\,$) obtained from the graph $G$ by deleting (respectively, contracting) an arbitrary edge $e$ from it. This work provides information about the hyperbolicity constant of minor graphs.

math.MG↗

arXiv2014

Hyperbolicity in the corona and join of graphs

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $δ$-\emph{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\ge 0: \, X \, \text{ is $δ$-hyperbolic}\,\}\,.$ Some previous works characterize the hyperbolic product graphs (for the Cartesian product, strong product and lexicographic product) in terms of properties of the factor graphs. In this paper we characterize the hyperbolic product graphs for graph join $G_1\uplus G_2$ and the corona $G_1\diamond G_2$: $G_1\uplus G_2$ is always hyperbolic, and $G_1\diamond G_2$ is hyperbolic if and only if $G_1$ is hyperbolic. Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join $G_1\uplus G_2$ and the corona $G_1\diamond G_2$.

math.CO↗

arXiv2014

Computing the alliance polynomial of a graph

The alliance polynomial of a graph $Γ$ with order $n$ and maximum degree $δ_1$ is the polynomial $A(Γ; x) = \sum_{k=-δ_1}^{δ_1} A_{k}(Γ) \, x^{n+k}$, where $A_{k}(Γ)$ is the number of exact defensive $k$-alliances in $Γ$. We provide an algorithm for computing the alliance polynomial. Furthermore, we obtain some properties of $A(Γ; x)$ and its coefficients. In particular, we prove that the path, cycle, complete and star graphs are characterized by their alliance polynomials. We also show that the alliance polynomial characterizes many graphs that are not distinguished by other usual polynomials of graphs.

math.CO↗

arXiv2014

New inequalities on the hyperbolicity constant of line graphs

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $δ$-\emph{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\ge 0: \, X \, \text{ is $δ$-hyperbolic}\,\}\,. $ The main result of this paper is the inequality $δ(G) \le δ(\mathcal L(G))$ for the line graph $\mathcal L(G)$ of every graph $G$. We prove also the upper bound $δ(\mathcal L(G)) \le 5 δ(G)+ 3 l_{max}$, where $l_{max}$ is the supremum of the lengths of the edges of $G$. Furthermore, if every edge of $G$ has length $k$, we obtain $δ(G) \le δ(\mathcal L(G)) \le 5 δ(G)+ 5k/2$.

math.CO↗

arXiv2008

Sobolev spaces with respect to measures in curves and zeros of Sobolev orthogonal polynomials

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces.

math.FA↗