arXiv subjects

Walter Carballosa

Publications and source records attributed to Walter Carballosa.

At least 19 recordsLinked to original sources

arXiv2025

Encoding and Enumerating Acyclic Orientations of Graphs

In this work we study the acyclic orientations of graphs. We obtain an encoding of the acyclic orientations of the complete $p$-partite graph with size of its parts $n_1,n_2,\ldots,n_p$ via a vector with $p$ symbols and length $n=n_1+n_2+\ldots+n_p$ when the parts are fixed but not the vertices in each part. We also give a recursive way to construct all acyclic orientations of a complete multipartite graph, this construction can be done by computer easily in order $\mathcal{O}(n)$. Furthermore, we obtain a closed formula for non-isomorphic acyclic orientations of both the complete multipartite graphs and the complete multipartite graphs with a directed spanning tree. Moreover, we obtain a closed formula for the number of acyclic orientations of a complete multipartite graph $K_{n_1,\ldots,n_p}$ with labelled vertices. Finally, we obtain a way encode all acyclic orientations of an arbitrary graph as a permutation code. Using the codification mentioned above we obtain sharp upper and lower bounds of the number of acyclic orientations of a graph.

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arXiv2025

On the boundary polynomial of a graph

In this work, we introduce the boundary polynomial of a graph $G$ as the ordinary generating function in two variables $B(G;x,y):= \displaystyle\sum_{S\subseteq V(G)} x^{|B(S)|}y^{|S|}$, where $B(S)$ denotes the outer boundary of $S$. We investigate this graph polynomial obtaining some algebraic properties of the polynomial. We found that some parameters of $G$ are algebraically encoded in $B(G;x,y)$, \emph{e.g.}, domination number, Roman domination number, vertex connectivity, and differential of the graph $G$. Furthermore, we compute the boundary polynomial for some classic families of graphs. We also establish some relationships between $B(G;x,y)$ and $B(G^\prime;x,y)$ for the graphs $G^\prime$ obtained by removing, adding, and subdividing an edge from $G$. In addition, we prove that a graph $G$ has an isolated vertex if and only if its boundary polynomial has a factor ($y+1$). Finally, we show that the classes of complete, complete without one edge, empty, path, cycle, wheel, star, double-star graphs, and many others are characterized by the boundary polynomial.

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arXiv2023

Mathematical results on harmonic polynomials

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 polynomials. Also, we show that two graphs with the same harmonic polynomial have to be similar.

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arXiv2022

Total $k$-domination in Cartesian product of complete graphs

Let $G=(V,E)$ be a finite undirected graph. A set $S$ of vertices in $V$ is said to be total $k$-dominating if every vertex in $V$ is adjacent to at least $k$ vertices in $S$. The total $k$-domination number, $γ_{kt}(G)$, is the minimum cardinality of a total $k$-dominating set in $G$. In this work we study the total $k$-domination number of Cartesian product of two complete graphs which is a lower bound of the total $k$-domination number of Cartesian product of two graphs. We obtain new lower and upper bounds for the total $k$-domination number of Cartesian product of two complete graphs. Some asymptotic behaviors are obtained as a consequence of the bounds we found. In particular, we obtain that $\displaystyle\liminf_{n\to\infty}\frac{γ_{kt}(G\Box H)}{n}\leq 2\,\left(\left\lceil\frac{k}{2}\right\rceil^{-1}+\left\lfloor\frac{k+4}{2}\right\rfloor^{-1}\right)^{-1}$ for graphs $G,H$ with order at least $n$. We also prove that the equality is attained if and only if $k$ is even. The equality holds when $G,H$ are both isomorphic to the complete graph, $K_n$, with $n$ vertices. Furthermore, we obtain closed formulas for the total $2$-domination number of Cartesian product of two complete graphs of whatever order. Besides, we prove that, for $k=3$, the inequality above is improvable to $\displaystyle\liminf_{n\to\infty} γ_{3t}(K_n\Box K_n)/n \leq 11/5$.

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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 $\left\langle d \right\rangle$.

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arXiv2020

Relations between some topological indices and the line graph

The concepts of geometric-arithmetic and harmonic indices were introduced in the area of chemical graph theory recently. They have proven to correlate well with physical and chemical properties of some molecules. The aim of this paper is to obtain new inequalities involving the first Zagreb, the harmonic, and the geometric-arithmetic $GA_1$ indices. Furthermore, inequalities relating these indices and line graphs are proven.

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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.

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arXiv2018

Independence and matching numbers of some token graphs

Let $G$ be a graph of order $n$ and let $k\in\{1,\ldots,n-1\}$. The $k$-token graph $F_k(G)$ of $G$, is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. We study the independence and matching numbers of $F_k(G)$. We present a tight lower bound for the matching number of $F_k(G)$ for the case in which $G$ has either a perfect matching or an almost perfect matching. Also, we estimate the independence number for bipartite $k$-token graphs, and determine the exact value for some graphs.

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arXiv2016

Hyperbolicity of direct products 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, strong, join, corona and lexicographic products) in terms of properties of the factor graphs. However, the problem with the direct product is more complicated. In this paper, we prove that if the direct product $G_1\times G_2$ is hyperbolic, then one factor is hyperbolic and the other one is bounded. Also, we prove that this necessary condition is, in fact, a characterization in many cases. In other cases, we find characterizations which are not so simple. Furthermore, we obtain formulae or good bounds for the hyperbolicity constant of the direct product of some important graphs.

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arXiv2016

Regularity and Planarity of Token Graphs

Let $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper we characterize precisely, for each value of $k$, which graphs have a regular $k$-token graph and which connected graphs have a planar $k$-token graph.

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arXiv2016

Cohen-Macaulayness of triangular graphs

We study the Cohen-Macaulay property of triangular graphs $T_n$. We show that $T_2$, $T_3$ and $T_5$ are Cohen-Macaulay graphs, and that $T_4$, $T_6$, $T_8$ and $T_n$ are not Cohen-Macaulay graphs, for $n\geq 10$. Finally, we prove that over fields of characteristic zero $T_7$ and $T_9$ are Cohen-Macaulay.

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arXiv2015

Computing the strong alliance polynomial of a graph

We introduce the strong alliance polynomial of a graph. The strong alliance polynomial of a graph $G$ with order n and strong defensive alliance number $a(G)$ is the polynomial $a(G;x):=\sum_{i=a(G)}^{n}\, a_i(G)\ x^i$, where $a_{k}(G)$ is the number of strong defensive alliances with cardinality $k$ in $G$. We obtain some properties of $a(G; x)$ and its coefficients. In particular, we compute strong alliance polynomial for path, cycle, complete, start, complete bipartite and double star graphs; some of them verify unimodality.

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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 $10$.

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arXiv2015

Gromov hyperbolicity in lexicographic product 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}\}.$ In this paper we characterize the lexicographic product of two graphs $G_1\circ G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the lexicographic product graph $G_1\circ G_2$ is hyperbolic if and only if $G_1$ is hyperbolic, unless if $G_1$ is a trivial graph (the graph with a single vertex); if $G_1$ is trivial, then $G_1\circ G_2$ is hyperbolic if and only if $G_2$ is hyperbolic. In particular, we obtain the sharp inequalities $δ(G_1)\le δ(G_1\circ G_2) \le δ(G_1) + 3/2$ if $G_1$ is not a trivial graph, and we characterize the graphs for which the second inequality is attained.

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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.

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arXiv2015

Solving a conjecture about tessellation graphs of $\mathbb R^2$

In the paper Planarity and Hyperbolicity in Graphs, the authors present the following conjecture: every tessellation of the Euclidean plane with convex tiles induces a non-hyperbolic graph. It is natural to think that this statement holds since the Euclidean plane is non-hyperbolic. Furthermore, there are several results supporting this conjecture. However, this work shows that the conjecture is false.

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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 $G_1\diamond G_2$.

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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.

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