Papers by J. M. Sigarreta
Discussiones Mathematicae Graph Theory, 2007
In this paper we obtain several tight bounds on different types of alliance numbers of a graph, n... more In this paper we obtain several tight bounds on different types of alliance numbers of a graph, namely (global) defensive alliance number, global offensive alliance number and global dual alliance number. In particular, we investigate the relationship between the alliance numbers of a graph and its algebraic connectivity, its spectral radius, and its Laplacian spectral radius.
Ars Combinatoria Waterloo Then Winnipeg, Dec 31, 2015
The alliance polynomial of a graph Γ with order n and maximum degree δ1 is the polynomial A(Γ; x)... more The alliance polynomial of a graph Γ with order n and maximum degree δ1 is the polynomial A(Γ; x) = δ 1 k=−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.
Educación Matemática, 2021
In this paper we present a strategy for the study of the teaching meanings that teachers have reg... more In this paper we present a strategy for the study of the teaching meanings that teachers have regarding the concept of slope. It consists of three stages: the elaboration of empirical instruments for the collection of information, the implementation of the instruments, and the analysis of the information collected to identify, classify and evaluate the teachers' meanings. This strategy explores the coherence between what the teacher intends to convey, what he imagines he conveys, and what his interlocutor (the student) conceives to be conveyed. In order to analyze the structural and functional quality of the instrumental component of the strategy, a study of its evaluation before a panel of experts is presented. The results are processed with the help of a technique for the representation of the ordering by similarity to the ideal solution (TOPSIS), based on fuzzy data.
Mathematical Biosciences and Engineering
The aim of this work is to obtain new inequalities for the variable symmetric division deg index ... more 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 the degree of $ u $, and $ \alpha \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_\alpha(G) $ index on random graphs and we demonstrate that the ratio $ \langle SDD_\alpha(G) \rangle/n $ ($ n $ is the order of the graph) depends only on the average degree $ \langle d \rangle $.
1 Facultad de Ciencias Qúımicas, Benemérita Universidad Autónoma de Puebla, Puebla 72570, Mexico ... more 1 Facultad de Ciencias Qúımicas, Benemérita Universidad Autónoma de Puebla, Puebla 72570, Mexico 2Instituto de F́ısica, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico 3Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain 4Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, Acapulco Gro. 39650, Mexico [email protected], [email protected], [email protected], [email protected]
Symmetry, 2018
A graph operator is a mapping F : Γ → Γ ′ , where Γ and Γ ′ are families of graphs. The different... more A graph operator is a mapping F : Γ → Γ ′ , where Γ and Γ ′ are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Λ ( G ) ; subdivision graph, S ( G ) ; total graph, T ( G ) ; and the operators R ( G ) and Q ( G ) . In particular, we get relationships such as δ ( G ) ≤ δ ( R ( G ) ) ≤ δ ( G ) + 1 / 2 , δ ( Λ ( G ) ) ≤ δ ( Q ( G ) ) ≤ δ ( Λ ( G ) ) + 1 / 2 , δ ( S ( G ) ) ≤ 2 δ ( R ( G ) ) ≤ δ ( S ( G ) ) + 1 and δ ( R ( G ) ) − 1 / 2 ≤ δ ( Λ ( G ) ) ≤ 5 δ ( R ( G ) ) + 5 / 2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.
Discrete Applied Mathematics, 2009
Let Γ = (V, E) be a simple graph. For a nonempty set X ⊆ V , and a vertex v ∈ V , δ X (v) denotes... more Let Γ = (V, E) be a simple graph. For a nonempty set X ⊆ V , and a vertex v ∈ V , δ X (v) denotes the number of neighbors v has in X. A nonempty set S ⊆ V is a defensive k-alliance in Γ = (V, E) if δ S (v) ≥ δS(v) + k, ∀v ∈ S. A defensive k-alliance S is called global if it forms a dominating set. The global defensive k-alliance number of Γ, denoted by γ a k (Γ), is the minimum cardinality of a defensive k-alliance in Γ. We study the mathematical properties of γ a k (Γ).
Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in th... more 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 for the hyperbolicity constant of (finite and infinite) circular-arc graphs. Moreover, we obtain bounds for the hyperbolicity constant of the complement and line of any circular-arc graph. In order to do that, we obtain new results about regular, chordal and line graphs which are interesting by themselves.
This paper identifies a specific contents-based strategy for problem solving based on analytical ... more This paper identifies a specific contents-based strategy for problem solving based on analytical geometry procedures. Here, an appropriate methodology for putting the strategy into practice, will be exposed.
Let $G=(V,E)$ be a simple graph. For a nonempty set $X\subset V,$ and a vertex $v\in V,$ $\delta_... more Let $G=(V,E)$ be a simple graph. For a nonempty set $X\subset V,$ and a vertex $v\in V,$ $\delta_{X}(v)$ denotes the number of neighbors $v$ has in $X.$ A nonempty set $S\subset V$ is an \emph{offensive $r$-alliance} in $G$ if $\delta_S(v)\ge \delta_{\bar{S}}(v)+r,$ $\forall v\in \partial (S),$ where $\partial (S)$ denotes the boundary of $S$. An offensive $r$-alliance $S$ is called \emph{global} if it forms a dominating set. The \emph{global offensive $r$-alliance number} of $G$, denoted by $\gamma_{r}^{o}(G)$, is the minimum cardinality of a global offensive $r$-alliance in $G$. We show that the problem of finding optimal (global) offensive $r$-alliances is NP-complete and we obtain several tight bounds on $\gamma_{r}^{o}(G)$.
arXiv: Combinatorics, 2013
The aim of this paper is to study some parameters of simple graphs related with the degree of the... more The aim of this paper is to study some parameters of simple graphs related with the degree of the vertices. So, our main tool is the $n\times n$ matrix ${\cal A}$ whose ($i,j$)-entry is $$ a_{ij}= \left\lbrace \begin{array}{ll} \frac{1}{\sqrt{\delta_i\delta_j}} & {\rm if }\quad v_i\sim v_j ; \\ 0 & {\rm otherwise,} \end{array} \right. $$ where $\delta_i$ denotes the degree of the vertex $v_i$. We study the Randi\'{c} index and some interesting particular cases of conditional excess, conditional Wiener index, and conditional diameter. In particular, using the matrix ${\cal A}$ or its eigenvalues, we obtain tight bounds on the studied parameters.
In this paper we obtain several tight bounds on different types of alliance numbers of a graph, n... more In this paper we obtain several tight bounds on different types of alliance numbers of a graph, namely (global) defensive alliance number, global offensive alliance number and global dual alliance number. In particular, we investigate the relationship between the alliance numbers of a graph and its algebraic connectivity, its spectral radius, and its Laplacian spectral radius.
Let $G=(V,E)$ be a simple connected graph and $d_i$ be the degree of its $i$th vertex. In a recen... more Let $G=(V,E)$ be a simple connected graph and $d_i$ be the degree of its $i$th vertex. In a recent paper [J. Math. Chem. 46 (2009) 1369-1376] the first geometric-arithmetic index of a graph $G$ was defined as $$GA_1=\sum_{ij\in E}\frac{2 \sqrt{d_i d_j}}{d_i + d_j}.$$ This graph invariant is useful for chemical proposes. The main use of $GA_1$ is for designing so-called quantitative structure-activity relations and quantitative structure-property relations. In this paper we obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize the graphs which make the inequalities tight. In particular, we improve some known results, generalize other, and we relate $GA_1$ to other well-known topological indices.
Este articulo aborda la evolucion de la resolucion de problemas matematicos desde una perspectiva... more Este articulo aborda la evolucion de la resolucion de problemas matematicos desde una perspectiva historico-didactica, tomando como guia cuatro etapas fundamentales: la Antiguedad, partiendo desde el 2000 a. n. e hasta la caida del Imperio Romano en el siglo V n. e; se sigue con la Edad Media, hasta el siglo XV; luego la Era Moderna, que finaliza con la alborada del siglo XX; y se concluye en la epoca Contemporanea.
In this paper we study mathematical properties of alliances (defensive alliances, offensive allia... more In this paper we study mathematical properties of alliances (defensive alliances, offensive alliances and dual alliances) in planar graphs. In particular, we obtain several tight bounds on different types of alliance numbers of a planar graph.
Let Γ = (V,E) be a simple graph. For a nonempty set X ⊆ V , and a vertex v ∈ V , δX(v) denotes th... more Let Γ = (V,E) be a simple graph. For a nonempty set X ⊆ V , and a vertex v ∈ V , δX(v) denotes the number of neighbors v has in X. A nonempty set S ⊆ V is a defensive k-alliance in Γ = (V,E) if δS(v) ≥ δS̄(v)+k, ∀v ∈ S. A defensive k-alliance S is called global if it forms a dominating set. The global defensive k-alliance number of Γ, denoted by γ k(Γ), is the minimum cardinality of a defensive k-alliance in Γ. We study the mathematical properties of γ k(Γ).
arXiv: Combinatorics, 2006
A \emph{defensive} (\emph{offensive}) $k$-\emph{alliance} in $\Gamma=(V,E)$ is a set $S\subseteq ... more A \emph{defensive} (\emph{offensive}) $k$-\emph{alliance} in $\Gamma=(V,E)$ is a set $S\subseteq V$ such that for every $v\in S$ ($v\in \partial S$), the number of neighbors $v$ has in $S$ is at least $k$ more than the number of neighbors it has in $V\setminus S$. A set $X\subseteq V$ is \emph{defensive} (\emph{offensive}) $k$-\emph{alliance free,} if for all defensive (offensive) $k$-alliance $S$, $S\setminus X\neq\emptyset$, i.e., $X$ do not contain any defensive (offensive) $k$-alliance as a subset. A set $Y \subseteq V$ is a \emph{defensive} (\emph{offensive}) $k$-\emph{alliance cover}, if for all defensive (offensive) $k$-alliance $S$, $S\cap Y\neq\emptyset$, i.e., $Y$ contains at least one vertex from each defensive (offensive) $k$-alliance of $\Gamma$. In this paper we obtain several mathematical properties of defensive (offensive) $k$-alliances, $k$-alliance free sets and $k$-alliance cover sets, and we explore some of their interrelations.
In this work we obtain new lower and upper optimal bounds of general Sombor indices. Specifically... more 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.
Aequationes mathematicae, 2014
If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X, a geodesic triangle T = {x 1 , x 2 , x 3... more If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ 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 ≥ 0 : X 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 ⊎ G 2 and the corona G 1 ⋄ G 2 : G 1 ⊎ G 2 is always hyperbolic, and G 1 ⋄ 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 ⊎G 2 and the corona G 1 ⋄ G 2 .
Electronic Notes in Discrete Mathematics, 2014
ABSTRACT For an ordered subset $S = \{s_1, s_2,\dots s_k\}$ of vertices and a vertex $u$ in a con... more ABSTRACT For an ordered subset $S = \{s_1, s_2,\dots s_k\}$ of vertices and a vertex $u$ in a connected graph $G$, the metric representation of $u$ with respect to $S$ is the ordered $k$-tuple $ r(u|S)=(d_G(v,s_1), d_G(v,s_2),\dots,$ $d_G(v,s_k))$, where $d_G(x,y)$ represents the distance between the vertices $x$ and $y$. The set $S$ is a metric generator for $G$ if every two different vertices of $G$ have distinct metric representations. A minimum metric generator is called a metric basis for $G$ and its cardinality, $dim(G)$, the metric dimension of $G$. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae and tight bounds for the metric dimension of strong product graphs.
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Papers by J. M. Sigarreta