Riordan graphs I: Structural properties

Cheon, Gi-Sang; Jung, Ji-Hwan; Kitaev, Sergey; Mojallal, Seyed Ahmad

doi:10.1016/j.laa.2019.05.033

Cited by 17 publications

(33 citation statements)

References 26 publications

(58 reference statements)

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“…The notion of a Riordan graph was introduced in [6]. A simple labelled graph G with n vertices is a Riordan graph of order n if the adjacency matrix of G is an n × n symmetric (0, 1)-matrix given by A(G) = B(zg, f ) n + B(zg, f ) T n for some Riordan matrix (g, f ) over Z.…”

Section: Riordan Graphsmentioning

confidence: 99%

Word-representability of Toeplitz graphs

Cheon

Kim

et al. 2019

Discrete Applied Mathematics

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Distinct letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word of the form xyxy · · · (of even or odd length) or a word of the form yxyx · · · (of even or odd length). A graph G = (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy is an edge in E.In this paper we initiate the study of word-representable Toeplitz graphs, which are Riordan graphs of the Appell type. We prove that several general classes of Toeplitz graphs are word-representable, and we also provide a way to construct nonword-representable Toeplitz graphs. Our work not only merges the theories of Riordan matrices and word-representable graphs via the notion of a Riordan graph, but also it provides the first systematic study of word-representability of graphs defined via patterns in adjacency matrices. Moreover, our paper introduces the notion of an infinite word-representable Riordan graph and gives several general examples of such graphs. It is the first time in the literature when the word-representability of infinite graphs is discussed.

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Section: Riordan Graphsmentioning

confidence: 99%

Word-representability of Toeplitz graphs

Cheon

Kim

et al. 2019

Discrete Applied Mathematics

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“…Riordan graphs are proved to have a number of interesting (fractal) properties [3], which can be useful, e.g. in creating computer networks [9] with certain desirable features, such as • the design is to be simple and recursive;…”

Section: Introductionmentioning

confidence: 99%

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“…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 [3] 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 [3] 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.…”

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confidence: 99%

“…In this paper, we use a number of results in [3] 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.…”

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Riordan graphs II: Spectral properties

Cheon¹,

Jung²,

Kitaev³

et al. 2019

Linear Algebra and its Applications

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The authors of this paper have used the theory of Riordan matrices to introduce the notion of a Riordan graph in [3]. 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 [3] 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 [3] 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.

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Encoding Labelled p-Riordan Graphs by Words and Pattern-Avoiding Permutations

Iamthong

Jung

Kitaev

2020

Graphs and Combinatorics

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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.

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Riordan graphs I: Structural properties

Cited by 17 publications

References 26 publications

Word-representability of Toeplitz graphs

Word-representability of Toeplitz graphs

Riordan graphs II: Spectral properties

Encoding Labelled p-Riordan Graphs by Words and Pattern-Avoiding Permutations

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