×

Asymptotic spectral distributions of Manhattan products of \(C_{n}\sharp P_{m}\). (English) Zbl 1327.05137

Summary: The Manhattan product of directed cycles \(C_{n}\) and directed paths \(P_{m}\) is a diagraph. Recently, in quantum probability theory, several authors have studied the spectrum of a graph, as mentioned also by A. Hora and N. Obata [Quantum probability and spectral analysis of graphs. With aforeword by Professor Luigi Accardi. Berlin: Springer (2007; Zbl 1141.81005)]. In the paper, we study asymptotic spectral distribution of the Manhattan products of simple digraphs \(C_{n}\sharp P_{m}\). The limit of the spectral distribution of \(C_{n}\sharp P_{2}\) as \(n\to \infty \) exists in the sense of weak convergence, and its concrete form is obtained. We insist on the fact that this note does not contain any new results, it shows only some parallel results to [N. Obata, Interdiscip. Inf. Sci. 18, No. 1, 43–54 (2012; Zbl 1260.05137); Ann. Funct. Anal. 3, No. 2, 135–143 (2012; Zbl 1252.05129)]. But, we have only written this paper to convey the information from quantum probability to spectral analysis of graph.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C76 Graph operations (line graphs, products, etc.)
62E20 Asymptotic distribution theory in statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lee, H.-O.: Product Structures of Networks and Their Spectra. Master Thesis, Tohoku University (2011) · Zbl 1223.41017
[2] Obata, N.: Spectra of Manhattan products of directed paths \[P_n\sharp P_2\] Pn♯P2. Interdiscip. Inf. Sci. 18(1), 43-54 (2012) · Zbl 1260.05137
[3] Obata, N.: Manhattan products of digraphs: characteristic polynomials and examples. Ann. Funct. Anal. 3, 136-144 (2012) · Zbl 1252.05129
[4] Brualdi, R.A.: Spectra of digraphs. Linear Algebra Appl. 432, 2181-2213 (2010) · Zbl 1221.05177
[5] Comellas, F., Dalfo, C., Fiol, M.A., Mitjana, M.: The spectra of Manhattan street networks. Linear Algebra Appl. 429, 1823-1839 (2008) · Zbl 1144.05316
[6] Obata, N.: Notions of independence in quantum probability and spectral analysis of graphs. Am. Math. Soc. Transl. 223, 115-136 (2008) · Zbl 1170.46056
[7] Hora, A., Obata, N.: Quantum Probability and Spectral Analysis of Graphs. Springer, Berlin (2007) · Zbl 1141.81005
[8] Comellas, F., Dalfo, C., Fiol, M.A., Mitjana, M.: A spectral study of the Manhattan networks. Electron. Notes Discret. Math. 29, 267-271 (2007) · Zbl 1341.05151
[9] Cvetkovic, D.M.: Recent Results in the Theory of Graph Spectra. Elsevier, Amsterdam (1988) · Zbl 0634.05054
[10] Cvetkovic, D.M., Doob, M., Sachs, H.: Spectra of Graphs. Academic Press, New York (1979) · Zbl 0824.05046
[11] Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, London (1978) · Zbl 0389.33008
[12] Cvetkovic, D.M., Rowlinson, P., Simic, S.: Eigenspaces of Graphs. Cambridge University Press, Cambridge (2008) · Zbl 1143.05052
[13] Brualdi, R.A.: Spectra of digraphs. Linear Algebra Appl. 432, 2181-2213 (2010) · Zbl 1221.05177
[14] Zhang, F.: Matrix Theory: Basic Results and Techniques. Springer, Berlin (2011) · Zbl 1229.15002
[15] Jafarizadeh, M.A., Salimi, S.: Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix. Ann. Phys. 322, 1005-1033 (2007) · Zbl 1176.82012
[16] Deañoa, A., Huybrechs, D., Kuijlaars, A.B.J.: Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature. J. Approx. Theory 162, 2202-2224 (2010) · Zbl 1223.41017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.