Diameter of io-decomposable Riordan graphs of the Bell type

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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“…Thus the induced subgraph of Ṽ is the complete graph K ⌈log 2 n⌉+1 . By (26) we obtain the desired result.…”

Section: Four Families Of Riordan Graphsmentioning

confidence: 76%

“…However, it was shown in [26] that Conjecture 1 is false, because its statement does not work for certain graphs G n and values of n. See [26] for the updated version of Conjecture 1. In either case, we conclude our paper with the following conjecture.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Riordan graphs I: Structural properties

Cheon

Jung

Kitaev

et al. 2019

Linear Algebra and its Applications

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“…Toeplitz graphs have been studied in [1,3,11]. A Riordan graph G n (g, tg) is said to be of the Bell type and it has been studied in [1,4]. The well-known Pascal graph…”

Section: Introductionmentioning

confidence: 99%

Encoding labelled $p$-Riordan graphs by words and pattern-avoiding permutations

Iamthong¹,

Jung²,

Kitaev³

2020

Preprint

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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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Diameter of io-decomposable Riordan graphs of the Bell type

Cited by 2 publications

References 10 publications

Riordan graphs I: Structural properties

Riordan graphs I: Structural properties

Encoding labelled $p$-Riordan graphs by words and pattern-avoiding permutations

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