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Given a positive integer $m$, the $m$-step competition graph of a digraph $D$, denoted by $C^m(D)$, has the same vertex set as $D$ and has an edge between vertices $u$ and $v$ if and only if there exists a vertex $w$ such that there exist directed walks of length $m$ from $u$ to $w$ and from $v$ to $w$, respectively. In this paper, we completely characterize the convergence of $\{C^m(D)\}_{m=1}^{\infty}$ for a multipartite tournament $D$ based on the last nontrivial strong component of $D$. Furthermore, not only do we determine the limit in the case of convergence, but also in the event of divergence, we specify how $C^m(D)$ changes periodically depending on the value of $m$. Our results extend the work of Jung et al. [On the limit of the sequence $\{C^m (D)\}_{m=1}^{\infty}$ for a multipartite tournament $D$. Discrete Appl. Math., 340:1--13, 2023] which addresses the case of the last strong component being nontrivial, thereby completing the convergence analysis of $\{C^m(D)\}_{m=1}^{\infty}$ for a multipartite tournament $D$. Our results can also be expressed in terms of matrix sequence $\{A^m(A^T)^m\}_{m=1}^{\infty}$ for the adjacency matrix $A$ of $D$ and this part is also covered in the text., Comment: 22 pages, 6 figures. arXiv admin note: text overlap with arXiv:2211.10978

For an integer $k \ge 2$, let $A$ be a Boolean block matrix with blocks $A_{ij}$ for $1 \le i,j \le k$ such that $A_{ii}$ is a zero matrix and $A_{ij}+A_{ji}^T$ is a matrix with all elements $1$ but not both corresponding elements of $A_{ij}$ and $A_{ji}^T$ equal to $1$ for $i \neq j$. Jung~{\em et al.} [Competition periods of multipartite tournaments. {\it Linear and Multilinear Algebra}, https://doi.org/10.1080/03081087.2022.2038057] studied the matrix sequence $\{A^m(A^T)^m\}_{m=1}^{\infty}$. This paper, which is a natural extension of the above paper and was initiated by the observation that $\{A^m(A^T)^m\}_{m=1}^{\infty}$ converges if $A$ has no zero rows, computes the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^{\infty}$ if $A$ has no zero rows. To this end, we take a graph theoretical approach: noting that $A$ is the adjacency matrix of a multipartite tournament $D$, we compute the limit of the graph sequence $\left\{ C^m(D) \right\}_{m=1}^{\infty}$ when $D$ has no sinks.

In this paper, we study structural properties of Toeplitz graphs. We characterize $K_q$-free Toeplitz graphs for an integer $q \ge 3$ and give equivalent conditions for a Toeplitz graph $G_n\langle t_1, t_2,\ldots, t_k\rangle$ with $t_1<\cdots

In this paper, we compute competition indices and periods of multipartite tournaments. We first show that the competition period of an acyclic digraph $D$ is one and $\zeta(D) +1$ is a sharp upper bound of the competition index of $D$ where $\zeta(D)$ is the sink elimination index of $D$. Then we prove that, especially, for an acyclic $k$-partite tournament $D$, the competition index of $D$ is $\zeta(D)$ or $\zeta(D) +1$ for an integer $k \ge 3$. By developing useful tools to create infinitely many directed walks in a certain regular pattern from given directed walks, we show that the competition period of a multipartite tournament with sinks and directed cycles is at most three. We also prove that the competition index of a primitive digraph does not exceed its exponent., Comment: 20 pages

In this paper, we use the theory of Riordan matrices to introduce the notion of an oriented Riordan graph. The oriented Riordan graphs are a far-reaching generalization of the well known and well studied Toeplitz oriented graphs and tournament. The main focus in this paper is the study of structural properties of the oriented Riordan graphs which includes a fundamental decomposition theorem and fractal property. Finally, we introduce the generalization of the oriented Riordan graph who is called a $p$-Riordan graph., Comment: 18 pages, 3 figures. arXiv admin note: text overlap with arXiv:1710.04604

The notion of a $p$-Riordan graph generalizes that of a Riordan graph, which, in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. In this paper we introduce the notion of a $p$-Riordan word, and show how to encode $p$-Riordan graphs by $p$-Riordan words. For special important cases of Riordan graphs (the case $p=2$) and oriented Riordan graphs (the case $p=3$) we provide alternative encodings in terms of pattern-avoiding permutations and certain balanced words, respectively. As a bi-product of our studies, we provide an alternative proof of a known enumerative result on closed walks in the cube., Comment: To appear in Graphs and Combinatorics, 14 pages, 1 fiugure

The notion of a Riordan graph was introduced recently, and it is a far-reaching generalization of the well-known Pascal graphs and Toeplitz graphs. However, apart from a certain subclass of Toeplitz graphs, nothing was known on independent sets in Riordan graphs. In this paper, we give exact enumeration and lower and upper bounds for the number of independent sets for various classes of Riordan graphs. Remarkably, we offer a variety of methods to solve the problems that range from the structural decomposition theorem to methods in combinatorics on words. Some of our results are valid for any graph., Comment: Accepted for publication in Discrete Mathematics

Recently, in the paper \cite{CJKM1} we suggested the two conjectures about the diameter of io-decomposable Riordan graphs of the Bell type. In this paper, we give a counterexample for the first conjecture. Then we prove that the first conjecture is true for the graphs of some particular size and propose a new conjecture. Finally, we show that the second conjecture is true for some special io-decomposable Riordan graphs., Comment: 13 pages, 1 figure, 2 tables

The authors of this paper have used the theory of Riordan matrices to introduce the notion of a Riordan graph in \cite{CJKM}. Riordan graphs are proved to have a number of interesting (fractal) properties, and they are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The main focus in \cite{CJKM} is the study of structural properties of families of Riordan graphs obtained from certain infinite Riordan graphs. In this paper, we use a number of results in~\cite{CJKM} to study spectral properties of Riordan graphs. Our studies include, but are not limited to the spectral graph invariants for Riordan graphs such as the adjacency eigenvalues, (signless) Laplacian eigenvalues, nullity, positive and negative inertias, and rank. We also study determinants of Riordan graphs, in particular, giving results about determinants of Catalan graphs., Comment: 33 pages

In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desirable features, or in obtaining useful information when designing algorithms to compute values of graph invariants. The main focus in this paper is the study of structural properties of families of Riordan graphs obtained from infinite Riordan graphs, which includes a fundamental decomposition theorem and certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle. We will study spectral properties of the Riordan graphs in a follow up paper., Comment: 45 pages, 12 figures