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September 2003 - April 2016
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Publications (62)
Let D be a digraph with vertex set V(D) and arc set A(D). For a real function f defined on nonnegative real numbers, the vertex-degree function index \(H_{f}(D)\) of the digraph D is defined as
where \(d_u^+\) and \(d_u^-\) denote the outdegree and the indegree of u, respectively. In this paper we find the extremal values of \(H_{f}\) among orienta...
Las limitaciones y dificultades que presentan los estudiantes en el aprendizaje de la Matemática ponen de manifiesto la necesidad de dotar a los docentes de estrategias para favorecer su proceso de enseñanza-aprendizaje. En esa dirección, se propone una estrategia didáctica para aprovechar las potencialidades de la Representacion Gráfica como recur...
A dominating set in a graph is a set of vertices such that every vertex outside the set is adjacent to a vertex in the set. The domination number is the minimum cardinality of a dominating set in the graph. The problem of finding the minimum dominating set is a combinatorial optimization problem that has been proved to be NP-hard. Given the difficu...
Finding minimum dominating sets in graphs is a problem that has been widely studied in the literature. However, due to the increase in the size and complexity of networks, new algorithms with the ability to provide high quality solutions in short computing times are desirable. This work presents a Greedy Randomized Adaptive Search Procedure for dea...
Given a graph G with vertex set V(G) and edge set E(G), the geometric-arithmetic index is the valueGA(G)=∑uv∈E(G)2dudvdu+dv, where du and dv denote the degrees of the vertices u,v∈V(G), respectively. In this work we present an upper bound for the geometric-arithmetic index of trees in terms of the order and the domination number, and we characteriz...
Let G be a graph with vertex set V and edge set E, a set D⊆V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\subseteq V$$\end{document} is a total dominating set if ev...
Let G=(V,E) be a simple graph with vertex set V and edge set E. Let D be a subset of V, and let B(D) be the set of neighbors of D in V∖D. The differential ∂(D) of D is defined as |B(D)|−|D|. The maximum value of ∂(D) taken over all subsets D⊆V is the differential of G, denoted by ∂(G). The line graph L(G) of G=(V,E) is the graph whose vertex set is...
Let D be a digraph with arc set A(D). A vertex-degree-based topological index φ is defined in D asφ(D)=12∑uv∈A(D)φdu+,dv−,where du+ is the outdegree of vertex u, dv− is the indegree of vertex v, and φx,y is a (symmetric) function. We study in this paper the extremal value problem of a VDB topological index φ over the set of orientations of a tree T...
In this paper we study the $ k $-domination and total $ k $-domination numbers of catacondensed hexagonal systems. More precisely, we give the value of the total domination number, we find upper and lower bounds for the $ 2 $-domination number and the total $ 2 $-domination number, characterizing the catacondensed hexagonal systems which attain the...
Given two types of graph theoretical parameters ρ and σ, we say that a graph G is (σ,ρ)-perfect if σ(H)=ρ(H) for every non-trivial connected induced subgraph H of G. In this work we characterize (γw,τ)-perfect graphs, (γw,α′)-perfect graphs, and (α′,τ)-perfect graphs, where γw(G), τ(G) and α′(G) denote the weakly connected domination number, the ve...
Let G=(V,E) be a graph, and let β∈R. Motivated by a service coverage maximization problem with limited resources, we study the β-differential of G. The β-differential of G, denoted by ∂β(G), is defined as ∂β(G):=max{|B(S)|−β|S|suchthatS⊆V}. The case in which β=1 is known as the differential of G, and hence ∂β(G) can be considered as a generalizatio...
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...
Let D be a digraph with set of vertices V(D) and set of arcs A(D). Recently, the Randić index of D was extended to digraphs as1 where and are the out‐degree and in‐degree of u and v, respectively. In this paper we study the extremal values of R over the set of all orientations of hexagonal chains with k hexagons. Directed complex networks play an i...
In this paper we answer all the conjectures about the domination number of a catacondensed hexagonal system given in Automated Conjecturing VI: Domination Number of Benzenoids. Moreover, we give lower bounds for the domination number in catacondensed hexagonal systems using the number of hexagons and the number of branching hexagons.
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.
MSC: 05C05 05C35 05C69 Keywords: Randí c index Domination number a b s t r a c t The Randí c index is the topological index most widely used in applications for chemistry and pharmacology. It is defined for a graph G with vertex set V (G) and edge set E (G) as R (G) = u v ∈ E (G) 1 deg (u) deg (v) , where deg (u) and deg (v) denote the degrees of t...
In this paper we give bounds for the domination number in hexagonal chains and the exact value of this parameter for some particular hexagonal chains. We also find the hexagonal chains with minimum and maximum domination number, among all hexagonal chains with a fixed number of hexagons.
The Hermite–Hadamard inequality was first considered for convex
functions and has been studied extensively. Recently, many extensions were
given with the use of general convex functions. In this paper we present some
variants of the Hermite–Hadamard inequality for general convex functions in the
context of q-calculus. From our theorems, we deduce s...
Let G n be the set of graphs with n non-isolated vertices. In this paper we identify vertex–degree–based topological indices over G n with vectors in R h , the Euclidean space with h = ⁽ n ⁻2¹⁾ⁿ coordinates. In this setting, we give an interpretation of the extremal values of a topological index in terms of angles between vectors in R h . Then we c...
A subset D of vertices of a graph G is a global total k-dominating set if D is a totalk-dominating set of both G and G¯. The global total k-domination number of G is the minimum cardinality of a global total k-dominating set of G and it is denoted by γktg(G). In this paper we introduce this concept and we begin the study of its mathematical propert...
Let G=(V,E) be a graph, a set S⊆V is a total k-dominating set if every vertex v∈V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper, we obtain several tight bounds for the total k-domination number of the strong product of two graphs. In particular, we inv...
Let G = (V, E) be a graph; a set S ⊆ V is a total k-dominating set if every vertex v ϵ V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of a graph. In particular, we investigate the relati...
Let G = ( V , E ) be a simple graph with vertex set V and edge set E. Let D be a subset of V, and let B ( D ) be the set of neighbours of D in V ∖ D . The differential ∂ ( D ) of D is defined as | B ( D ) | − | D | . The maximum value of ∂ ( D ) taken over all subsets D ⊆ V is the differential ∂ ( G ) of G. For β ∈ ( − 1 , Δ ) , the β-differential...
Let G=(V,E) be a graph of order n and let B(D) be the set of vertices in V(set minus)D that have a neighbor in the vertex set D. The differential of a vertex set D is defined as ∂(D)=|B(D)|-|D| and the maximum value of ∂(D) for any subset D of V is the differential of G, denoted by ∂(G). A Roman dominating function of G is a function f:V→(0,1,2) su...
Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a total k-dominating set if every vertex \(v\in V\) has at least k neighbors in S. The total k-domination number \(\gamma _{kt}(G)\) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of the Cartesian produ...
If X is a geodesic metric space and 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 the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the other two sides, for every geodesic triangle T in X. An important problem...
If is a geodesic metric space and , a geodesic triangle is the union of the three geodesics , and in . The space is -hyperbolic (in the Gromov sense) if any side of is contained in a -neighborhood of the union of the two other sides, for every geodesic triangle in . We denote by the sharpest hyperbolicity constant of , i.e., . In the study of any p...
Let Gamma(V, E) be a graph of order n, and let B(S) be the set of vertices in V \ S that have a neighbor in the set S. The differential of a set S is defined as partial derivative(5) = vertical bar B(S)vertical bar - vertical bar S vertical bar, and the differential of a graph to equal the max{partial derivative(S)} for any subset S of V. In this p...
Let be a graph of order n and let be the set of vertices in that have a neighbor in the vertex set D. The differential of D is defined as and the differential of a graph G, written , is equal to . If G is connected and , is known. This immediately leads to a linear vertex kernel result (in the terminology of parameterized complexity) for the proble...
En este trabajo se realiza un estudio estadístico descriptivo de los resultados
que el alumnado en el Grado de Ingeniería Informática de la Universidad Pablo de
Olavide obtuvieron al resolver las cuestiones que les fueron planteadas en un aprueba
inicial, al comienzo del curso académico, donde se planteaban cuestiones para obtener
información sobre...
If X is a geodesic metric space and x1,x2,x3∈Xx1,x2,x3∈X, a geodesic triangle T={x1,x2,x3}T={x1,x2,x3} is the union of the three geodesics [x1x2],[x2x3][x1x2],[x2x3] and [x3x1][x3x1] 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 tr...
In this work we obtain several tight bounds for the differential of a graph and we relate the differential of a graph with some known parameters of the graph.
Let G=(V, E) be a graph of order n and let B(D) be the set of vertices in V ∖ D that have a neighbour in the set D. The differential of a set D is defined as ∂ (D)=|B(D)|−|D| and the differential of a graph to equal the maximum value of ∂(D) for any subset D of V. In this paper we obtain several tight bounds for the differential of strong product g...
We are studying computational complexity aspects of the differential of a graph, a graph parameter previously introduced to model ways of influencing a network. We obtain NP hardness results also for very special graph classes, such as split graphs and cubic graphs. This motivates to further classify this problem in terms of approximability. Here,...
Let G = (V,E) be a graph of order n and let B(S) be the set of vertices in V \ S that have a neighbor in the vertex set S. The differential of a vertex set S is defined as ∂(S) = |B(S)|-|S| and the maximum value of ∂(S) for any subset S of V is the differential of G. A Roman dominating function of G is a function f : V → {0, 1, 2} such that every v...
In this paper we prove that the study of the hyperbolicity on graphs can be reduced to the study of the hyperbolicity on simpler graphs. In particular, we prove that the study of the hyperbolicity on a graph with loops and multiple edges can be reduced to the study of the hyperbolicity in the same graph without its loops and multiple edges; we also...
Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\subseteq V$,
and vertex $v\in V$, let $\delta_S(v)=|\{u\in S:uv\in E\}|$ denote the
cardinality of the set of neighbors of $v$ in $S$, and let $\bar{S}=V-S$.
Consider the following condition: {equation}\label{alliancecondition}
\delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which st...
Let G=(V,E)G=(V,E) be a graph of order nn and let B(D)B(D) be the set of vertices in V∖DV∖D that have a neighbor in a set DD. The differential of a set DD is defined as ∂(D)=|B(D)|−|D|∂(D)=|B(D)|−|D| and the differential of a graph to equal the maximum value of ∂(D)∂(D) for any subset DD of VV. In this paper, we obtain several tight lower bounds fo...
Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\subseteq V$, and vertex $v\in V$, let $\delta_S(v)=|\{u\in S:uv\in E\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\bar{S}=V-S$. Consider the following condition: {equation}\label{alliancecondition} \delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which st...
If X is a geodesic metric space and 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 the Gromov sense) if any geodesic side of T is contained in a δ-neighborhood of the union of the two other geodesic sides, for every geodesic triangle T in X. We denote...
A defensive $k$-alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at least $k$ more neighbors in $S$ than it has outside of $S$. A defensive $k$-alliance $S$ is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive $k...
If X is a geodesic metric space and x1,x2,x3∈X, a geodesic triangleT={x1,x2,x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] 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. We denote by δ(X) the sharp h...
An offensive k-alliance in a graph is a set S of vertices with the property that every vertex in the boundary of S has at least k more neighbors in S than it has outside of S. An offensive k-alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) offensive...
We showed that computing the differential of a graph is NP-hard even on split graphs and on cubic graphs. Conversely, with a standard parameterization, the problem admits a kernel of linear order and is hence in FPT.
We investigate the relationship between global offensive k-alliances and some characteristic sets of a graph including r-dependent sets, τ-dominating sets and standard dominating sets. In addition, we discuss the close relationships that exist among the (global) offensive ki-alliance number of Γi,i∈{1,2}, and the (global) offensive k-alliance numbe...
In this paper we deal with Gromov hyperbolic graphs. We obtain several tight bounds for the hyperbolicity constant of a graph. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its edge number. We prove that the study of the hyperbolicity on graphs can be reduced to the study of the hyperbolicity on si...
We investigate the relationship between geodetic sets, k-geodetic sets, dominating sets and independent sets in arbitrary graphs. As a consequence of the study we provide several tight bounds on the geodetic number of a graph.
Let (T 1 , . . . , T N) be a N −tuple of positive operators with respect a Markushevich basis which are defined on a Hausdorff topological vector space. In this work we extend the notion of weak local quasinilpotence to N −tuples of operators (not-necessarily commuting). Under the hypothesis of existence of positive vectors, joint weak locally quas...
In this paper, we obtain several tight bounds of the defensive k-alliance number in the complement graph from other parameters of the graph. In particular, we investigate the relationship between the alliance numbers of the complement graph and the minimum and maximum degree, the domination number and the isoperimetric number of the graph. Moreover...
A defensive $k$-alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at least $k$ more neighbors in $S$ than it has outside of $S$. A defensive $k$-alliance $S$ is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive $k...
In this article we used the Klein group and the problems solving for study of two very general modes of reasoning and their relationship with the reversibility of thought processes.
Let be a completely non-unitary contraction having a 2×2 singular characteristic function Θ1; that is, Θ1=[θi,j]i,j=1,2 with θij∈H∞ and det(Θ1)=0. As it is well known, Θ1 is a singular matrix if and only if Θ1 can be written as where w1,m1,a1,b1,c1,d1∈H∞ are such that (i) w1 is an outer function with |w1|⩽1, (ii) m1 is an inner function, (iii) 2|a1...
Asymptotic estimates of the norms of orbits of certain operators that commute with the classical Volterra operator V acting on Lp[0,1], with 1⩽p⩽∞, are obtained. The results apply not only to the Riemann–Liouville operator Vr and to I+Vr with r>0, but also to operators of the form ϕ(V), where ϕ is a holomorphic function at zero. The method to obtai...
Let $T_1 \in \mathscr B( \mathscr H_1)$ be a completely non-unitary contraction having a non-zero characteristic function $\Theta_1$ which is a $2 \times 1$ column vector of functions in $H^\infty$. As it is well-known, such a function $\Theta_1$ can be written as $ \Theta_1=w_1 m_1 \left[ {a_1} \atop {b_1} \right] $ where $w_1, m_1, a_1, b_1 \in H...
Hankel operators and their symbols, as generalized by V. Pták and P. Vrbová, are considered. The present note provides a parametric
labeling of all the Hankel symbols of a given Hankel operator X by means of Schur class functions. The result includes uniqueness criteria and a Schur like formula. As a by-product, a new
proof of the existence of Hank...
A defensive (offensive) k-alliance in Γ = (V,E) is a set S ⊆ V such that every υ in S (in the boundary of S) has at least k more neighbors in S than it has in V / S. A set X ⊆ V is defensive (offensive) k-alliance free, if for all defensive (offensive) k-alliance S, S/X ≠ ∅, i.e., X does not contain any defensive (offensive) k-alliance as a subset....
A Toeplitz operator with respect to a contractive representation {Ts} of an abelian semigroup Σ in a Hilbert space H is an operator X ∈ B(H) such that X = TsXT * s for all s ∈ Σ. We show that if {Ts} has a minimal isometric dilation {Us} ⊂ B(K), then Toeplitz operators can be obtained in a unique way as compressions of operators Y ∈ B(K), called To...
A Toeplitz operator with respect to a contractive representation {Ts} of an abelian semigroup ∑ in a Hilbert space ℋ is an operator X ∈ B(ℋ) such that X = TsXTs* for all s ∈ ∑. We show that if {Ts} has a minimal isometric dilation {Us} ⊂ B{hooktop}(K{script}), then Toeplitz operators can be obtained in a unique way as compressions of operators Y ∈...