The spectrum of infinite regular line graphs

Shirai, Tomoyuki

doi:10.1090/s0002-9947-99-02497-6

Cited by 47 publications

(27 citation statements)

References 3 publications

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“…This is a version of spectral decimation, and uses an idea from [27] to control the L 2 norms of functions under spectral decimation. The second main abstract result is Theorem 3.1, which shows how to obtain the spectral resolution of the graph Laplacian on Γ 0 from the spectral resolution of the graph Laplacian on Γ using ideas from [34,40]. We note that the spectral resolution on Γ 0 may or may not contain the discrete eigenvalues equal to 6, and the explicit determination of the 6-eigenspace and its eigenprojector must be determined in a case-by-case manner.…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis on infinite Sierpiński fractafolds

Strichartz

Teplyaev

2012

JAMA

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Abstract. A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpiński gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpiński gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of spectral decimations to obtain explicit description of the eigenvalues and eigenfunctions. In this paper we combine the ideas from these earlier works to obtain a description of the spectral resolution of the Laplacian for noncompact fractafolds. Our main abstract results enable us to obtain a completely explicit description of the spectral resolution of the fractafold Laplacian. For some specific examples we turn the spectral resolution into a "Plancherel formula". We also present such a formula for the graph Laplacian on the 3-regular tree, which appears to be a new result of independent interest. In the end we discuss periodic fractafolds and fractal fields.

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Section: Introductionmentioning

confidence: 99%

Spectral analysis on infinite Sierpiński fractafolds

Strichartz

Teplyaev

2012

JAMA

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“…The subdivision graph S(G) is the graph that is obtain from G by replacing each of its edges by a path of length 2. The line graph L(G) of a graph G is the graph whose vertices are the edges of existing graph G, two vertices f and g are adjacent if and only if they are incident in G. Following [20], we can construct the line graph of subdivision graph L(S(G)) of a graph G, as follows:…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On topological properties of hexagonal and silicate networks

Akhter¹,

Imran²,

Farahani³

2017

HJMS

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Topological indices are the numerical quantities associated to a simple and finite graphs which represent its structure. There are certain types of topological indices such as degree based topological indices, distance based topological indices and counting related topological indices etc. The degree based topological indices are exploited to estimate the bioactivity of chemical compounds. In this paper, we compute first general Zagreb index, general Randić connectivity index, general sum-connectivity index, atom-bond connectivity index, geometricarithmetic index, ABC4 index and GA5 index of the line graphs of silicate networks, chain silicate networks and hexagonal networks by using the subdivision method.

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“…The stellated graph of G, denoted by st(G), is the line graph of the subdivision of G obtained from G by subdividing every edge once (see Figure 5). The stellated graph of G is also called inflated graph [4] or para-line graph of G [24]. A graph G is called a stellated graph if G = st(H) for some graph H. The spectrum of stellated graphs (or para-line graphs) have been studied in [24].…”

Section: Corollary 32 ([25] Cf[18]mentioning

confidence: 99%

Graph invertibility and median eigenvalues

Yang

Mandal

et al. 2017

Linear Algebra and its Applications

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Let (G, w) be a weighted graph with a weight-function w : E(G) → R\{0}. A weighted graph (G, w) is invertible to a new weighted graph if its adjacency matrix is invertible. A graph inverse has combinatorial interest and can be applied to bound median eigenvalues of a graph such as have physical meanings in Quatumn Chemistry. In this paper, we characterize the inverse of a weighted graph based on its Sachs subgraphs that are spanning subgraphs with only K2 or cycles (or loops) as components. The characterization can be used to find the inverse of a weighted graph based on its structures instead of its adjacency matrix. If a graph has its spectra split about the origin, i.e., half of eigenvalues are positive and half of them are negative, then its median eigenvalues can be bounded by estimating the largest and smallest eigenvalues of its inverse. We characterize graphs with a unique Sachs subgraph and prove that these graphs has their spectra split about the origin if they have a perfect matching. As applications, we show that the median eigenvalues of stellated graphs of trees and corona graphs belong to different halves of the interval [−1, 1].

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The spectrum of infinite regular line graphs

Cited by 47 publications

References 3 publications

Spectral analysis on infinite Sierpiński fractafolds

Spectral analysis on infinite Sierpiński fractafolds

On topological properties of hexagonal and silicate networks

Graph invertibility and median eigenvalues

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