Riordan graphs I: Structural properties

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

doi:10.48550/arxiv.1710.04604

Cited by 3 publications

(22 citation statements)

References 22 publications

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“…Throughout this paper, we write a ≡ b for a ≡ b (mod 2). Recently, we in [4,5] introduced a Riordan graph by using the notion of the Riordan matrix modulo 2 as follows. A simple labelled graph G with n vertices is a Riordan graph of order n if the adjacency matrix A(G) = [r i,j ] 1≤i,j≤n of G is an n×n symmetric (0, 1)-matrix given by A(G) ≡ (zg, f ) n + (zg, f ) T n , i.e.…”

Section: Introductionmentioning

confidence: 99%

“…In [4], we studied the structural properties of families of Riordan graphs obtained from infinite Riordan graphs, which include a fundamental decomposition theorem and certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle. A Riordan graph G n (g, f ) is called Bell type if f = zg.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 1.2. [4] Let G n = G n (g, f ) be a proper Riordan graph with the odd and even vertex sets…”

Section: Introductionmentioning

confidence: 99%

“…Lemma 1.3. [4] Let G n = G n (g, zg) be an io-decomposable Riordan graph of the Bell type. Then we have the following.…”

Section: Introductionmentioning

confidence: 99%

“…

Recently, in the paper [4] 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.

…”

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confidence: 99%

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Diameter of io-decomposable Riordan graphs of the Bell type

Jung¹

2019

Preprint

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Recently, in the paper [4] 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.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Definition 1.2. [4] Let G n = G n (g, f ) be a proper Riordan graph with the odd and even vertex sets…”

Section: Introductionmentioning

confidence: 99%

“…Lemma 1.3. [4] Let G n = G n (g, zg) be an io-decomposable Riordan graph of the Bell type. Then we have the following.…”

Section: Introductionmentioning

confidence: 99%

“…

Recently, in the paper [4] 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.

…”

mentioning

confidence: 99%

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Diameter of io-decomposable Riordan graphs of the Bell type

Jung¹

2019

Preprint

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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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Riordan graphs I: Structural properties

Cheon

Jung

Kitaev

et al. 2019

Linear Algebra and its Applications

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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.

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Riordan graphs I: Structural properties

Cited by 3 publications

References 22 publications

Diameter of io-decomposable Riordan graphs of the Bell type

Diameter of io-decomposable Riordan graphs of the Bell type

Riordan graphs II: Spectral properties

Riordan graphs I: Structural properties

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