Abstract
We are mainly concerned with sequences of graphs having a grid geometry,with a uniform local structure in a bounded domain Ω⊂Rd,d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega} {\\subset} \\mathbb{R}^{d}, d \\geq 1$$\\end{document}. When Ω=[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega = [0, 1]$$\\end{document}, such graphs include the standard Toeplitz graphs and, for Ω=[0,1]d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega = [0, 1]^{d}$$\\end{document},the considered class includes d-level Toeplitz graphs. In the general case, the underlyingsequence of adjacency matrices has a canonical eigenvalue distribution, inthe Weyl sense, and we show that we can associate to it a symbol f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{f}$$\\end{document}. The knowledgeof the symbol and of its basic analytical features provides many information onthe eigenvalue structure, of localization, spectral gap, clustering, and distributiontype.Few generalizations are also considered in connection with the notion of generalizedlocally Toeplitz sequences and applications are discussed, stemming e.g.from the approximation of differential operators via numerical schemes. Nevertheless,more applications can be taken into account, since the results presentedhere can be applied as well to study the spectral properties of adjacency matricesand Laplacian operators of general large graphs and networks.
Highlights
Spectral properties of the adjacency matrix and the Laplacian operator of graphs provide valuable insights regarding a large number of key features such as the Shannon capacity, Chromatic number, diameter, maximum cut, just to cite few of them, see [6, 35], which often play a central role in many applied real-world problems e.g. in physics and chemistry problems, see as references [15, 21, 34] and [12, Chapter 8]
The main result is the following: given a sequence of graphs having a grid geometry with a uniform local structure in a domain Ω, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and we show that we can associate to it a symbol function f
Assuming that the multi-index n = νn = (ν1n, ν2n, . . . , νdn) for a fixed ν ∈ Qd>0 := {(ν1, . . . , νd) ∈ Q : ν1, . . . , νd > 0}, it is immediate to see by the considerations above that nd−2An is the linear combination of graph-Laplacians of d-level diamond Toeplitz graphs with spectral distribution given by the following symbol function f : [−π, π]d → C2d×2d, with d k−1 d f(θ) = ck(ν) h(θr) ⊗ f ⊗
Summary
Spectral properties of the adjacency matrix and the Laplacian operator of graphs provide valuable insights regarding a large number of key features such as the Shannon capacity, Chromatic number, diameter, maximum cut, just to cite few of them, see [6, 35], which often play a central role in many applied real-world problems e.g. in physics and chemistry problems, see as references [15, 21, 34] and [12, Chapter 8]. The main result is the following: given a sequence of graphs having a grid geometry with a uniform local structure in a domain Ω, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense (see [5, 29] and references therein), and we show that we can associate to it a symbol function f.
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