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Luis Manuel Rivera

Publications and source records attributed to Luis Manuel Rivera.

17 recordsLinked to original sources
arXiv2025

Independence numbers of the 2-token graphs of some join graphs

The $2$-token graph $F_2(G)$ of a graph $G$ is the graph whose set of vertices consists of all the $2$-subsets of $V(G)$, where two vertices are adjacent if and only if their symmetric difference is an edge in $G$. Let $G$ be the join graph of $E_n$ and $H$, where $H$ is any graph. In this paper, we give a method to construct an independent set ${\mathcal I}'$ of $F_2(G)$ from an independent set ${\mathcal I}$ of $F_2(G)$ such that $|{\mathcal I}'| \geq |{\mathcal I}|$. As an application, we obtain the independence number of the $2$-token graphs of fan graphs $F_{n, m}$, wheel graphs $W_{n, m}$ and $E_n+K_n$.

math.CO↗
arXiv2023

The automorphism groups of some token graphs

In this paper we obtain the automorphism groups of the token graphs of some graphs. In particular we obtain the automorphism group of the $k$-token graph of the path graph $P_n$, for $n\neq 2k$. Also, we obtain the automorphism group of the $2$-token graph of the following graphs: cycle, star, fan and wheel graphs.

math.CO↗
arXiv2021

Hamiltonicity of the Double Vertex Graph and the Complete Double Vertex Graph of some Join Graphs

Let $G$ be a simple graph of order $n$. The double vertex graph $F_2(G)$ of $G$ is the graph whose vertices are the $2$-subsets of $V(G)$, where two vertices are adjacent in $F_2(G)$ if their symmetric difference is a pair of adjacent vertices in $G$. A generalization of this graph is the complete double vertex graph $M_2(G)$ of $G$, defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent vertices in $G$. In this paper we exhibit an infinite family of graphs (containing Hamiltonian and non-Hamiltonian graphs) for which $F_2(G)$ and $M_2(G)$ are Hamiltonian. This family of graphs is the set of join graphs $G=G_1 + G_2$, where $G_1$ and $G_2$ are of order $m\geq 1$ and $n\geq 2$, respectively, and $G_2$ has a Hamiltonian path. For this family of graphs, we show that if $m\leq 2n$ then $F_2(G)$ is Hamiltonian, and if $m\leq 2(n-1)$ then $M_2(G)$ is Hamiltonian.

math.CO↗
arXiv2021

Hamiltonicity of the Complete Double Vertex Graph of some Join Graphs

The complete double vertex graph $M_2(G)$ of $G$ is defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent vertices in $G$. In this paper we exhibit an infinite family of graphs $G$ (containing Hamiltonian and non-Hamiltonian graphs) for which $M_2(G)$ are Hamiltonian.

math.CO↗
arXiv2021

Hamiltonicity of Token Graphs of some Join Graphs

Let $G$ be a simple graph of order $n$ and let $k$ be an integer such that $1\leq k\leq n-1$. The $k$-token graph $G^{\{k\}}$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $G^{\{k\}}$ whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper we study the Hamiltonicity of the $k$-token graphs of some join graphs. As a consequence, we provide an infinite family of graphs (containing Hamiltonian and non-Hamiltonian graphs) for which their $k$-token graphs are Hamiltonian. Our result provides, to our knowledge, the first family of non-Hamiltonian graphs for which their $k$-token graphs are Hamiltonian, for $2<k<n-2$.

math.CO↗
arXiv2021

On the number of even roots of permutations

Let $σ$ be a permutation on $n$ letters. We say that a permutation $τ$ is an even (resp. odd) $k$th root of $σ$ if $τ^k=σ$ and $τ$ is an even (resp. odd) permutation. In this article, we obtain generating functions for the number of even and odd $k$th roots of permutations. Our result implies know generating functions of Moser and Wyman and also some generating functions for sequences in The On-line Encyclopedia of Integer Sequences (OEIS).

math.CO↗
arXiv2019

Independence numbers of some double vertex graphs and pair graphs

The combinatorial properties of double vertex graphs has been widely studied since the 90's. However only very few results are know about the independence number of such graphs. In this paper we obtain the independence numbers of the double vertex graphs of fan graphs and wheel graphs. Also we obtain the independence numbers of the pair graphs, that is a generalization of the double vertex graphs, of some families of graphs.

math.CO↗
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.

math.CO↗
arXiv2018

The packing number of the double vertex graph of the path graph

Neil Sloane showed that the problem of determine the maximum size of a binary code of constant weight 2 that can correct a single adjacent transposition is equivalent to finding the packing number of a certain graph. In this paper we solve this open problem by finding the packing number of the double vertex graph (2-token graph) of a path graph. This double vertex graph is isomorphic to the Sloane's graph. Our solution implies a conjecture of Rob Pratt about the ordinary generating function of sequence A085680.

math.CO↗
arXiv2017

Hamiltonicity of token graphs of fan graphs

In this note we show that the token graphs of fan graphs are Hamiltonian. This result provides another proof of the Hamiltonicity of Johnson graphs and also extends previous results obtained by Mirajkar and Priyanka Y. B.

math.CO↗
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.

math.CO↗
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.

math.AC↗
arXiv2015

Integer sequences and k-commuting permutations

Let $β$ be any permutation on $n$ symbols and let $c(k, β)$ be the number of permutations that $k$-commute with $β$. The cycle type of a permutation $β$ is a vector $(c_1, \dots, c_n)$ such that $β$ has exactly $c_i$ cycles of length $i$ in its disjoint cycle factorization. In this article we obtain formulas for $c(k, β)$, for some cycle types. We also express these formulas in terms of integer sequences as given in "The On-line Encyclopedia of Integer Sequences" (OEIS). For some of these sequences we obtain either new interpretations or relationships with sequences in the OEIS database.

math.CO↗
arXiv2014

Blocks in cycles and k-commuting permutations

Let $k$ be a nonnegative integer, and let $α$ and $β$ be two permutations of $n$ symbols. We say that $α$ and $β$ $k$-commute if $H(αβ, βα)=k$, where $H$ denotes the Hamming metric between permutations. In this paper, we consider the problem of finding the permutations that $k$-commute with a given permutation. Our main result is a characterization of permutations that $k$-commute with a given permutation $β$ in terms of blocks in cycles in the decomposition of $β$ as a product of disjoint cycles. Using this characterization, we provide formulas for the number of permutations that $k$-commute with a transposition, a fixed-point free involution and an $n$-cycle, for any $k$. Also, we determine the number of permutations that $k$-commute with any given permutation, for $k \leq 4$.

math.CO↗
arXiv2008

On low rank perturbation of matrices

The article is devoted to different aspects of the question "What can be done with a matrix by low rank perturbation?" It is proved that one can change a geometrically simple spectrum drastically by a rank 1 permutation, but the situation is quite different if one restricts oneself to normal matrices. Also, the Jordan normal form of a perturbed matrix is considered. It is proved that with respect to rank as a distance all almost unitary matrices are near unitary.

math.FA↗
arXiv2007

Almost solutions of equations in permutations

We will say that the permutations f_1,...,f_n is an e-solution of an equation if the normalized Hamming distance between its l.h.p. and r.h.p. is less than e. We give a sufficient conditions when near to an e-solution exists an exact solution and some examples when there does not exist such a solution.

math.GR↗