Hamiltonian properties of Toeplitz graphs

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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“…Using f ≡ tA(f ) it follows from Lemma 3.3 that if k = 2 s is an integer for s ≥ 0 then (15). Hence we complete the proof.…”

Section: Fractal Properties Of Riordan Graphsmentioning

confidence: 55%

“…j≤n is a lower triangular matrix with zero diagonal, if j + k(ℓ + 1) ≥ i then b i,j+k(ℓ+1) = 0 so that b i,j+km = 0 for all m ≥ ℓ + 2. Thus, if s satisfies(15) then b i+k,j+k ≡ b i,j ,i.e. (i, j) ∼ (i + 2 s , j + 2 s ).…”

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confidence: 99%

Riordan graphs I: Structural properties

Cheon

Jung

Kitaev

et al. 2019

Linear Algebra and its Applications

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“…Similarly to continuous cases, the discrete version of Laplacian A and corresponding Green matrices G play crucial roles. We focused our attention on fundamental and symmetric graphs such as complete graphs, cycles, platonic solids, truncated regular tetrahedrons, hexahedrons, octahedra and Toeplitz graphs [7]. We investigated a graph Laplacian and corresponding discrete Sobolev inequalities in a detailed manner.…”

Section: Introductionmentioning

confidence: 99%

A Hierarchical Structure for the Sharp Constants of Discrete Sobolev Inequalities on a Weighted Complete Graph

2017

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This paper clarifies the hierarchical structure of the sharp constants for the discrete Sobolev inequality on a weighted complete graph. To this end, we introduce a generalized-graph Laplacian A = I − B on the graph, and investigate two types of discrete Sobolev inequalities. The sharp constants C 0 (N; a) and C 0 (N) were calculated through the Green matrix G(a) = (A + aI) −1 (0 < a < ∞) and the pseudo-Green matrix G * = A † . The sharp constants are expressed in terms of the expansion coefficients of the characteristic polynomial of A. Based on this new discovery, we provide the first proof that each set of the sharp constants {C 0 (n; a)} N n=2 and {C 0 (n)} N n=2 satisfies a certain hierarchical structure.

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“…The band condition means that the edges can only join vertices located at a bounded distance from each other along the perimeter of the polygon. Töeplitz graphs are trickier to describe, see [8], roughly they are obtained from circulants by removing a few vertices and all adjacent edges so that the polygon opens up. Most of the recent studies of the graph energy of circulants use number theoretic methods, our work complements them with analytic and probabilistic approaches.…”

Section: Introductionmentioning

confidence: 99%

The asymptotic trace norm of random circulants and the graph energy

Koshkin

2016

Discuss. Math. Probability and Statistics

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We compute the expected normalized trace norm (matrix/graph energy) of random symmetric band circulant matrices and graphs in the limit of large sizes, and obtain explicit bounds on the rate of convergence to the limit, and on the probabilities of large deviations. We also show that random symmetric band Töeplitz matrices have the same limit norm assuming that their band widths remain small relative to their sizes. We compare the limit norms across a range of related random matrix and graph ensembles.

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Hamiltonian properties of Toeplitz graphs

Cited by 34 publications

References 7 publications

Riordan graphs I: Structural properties

Riordan graphs I: Structural properties

A Hierarchical Structure for the Sharp Constants of Discrete Sobolev Inequalities on a Weighted Complete Graph

The asymptotic trace norm of random circulants and the graph energy

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