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Transactions of the American Mathematical Society

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The spectrum of infinite regular line graphs


Author: Tomoyuki Shirai
Journal: Trans. Amer. Math. Soc. 352 (2000), 115-132
MSC (1991): Primary 39A12; Secondary 39A70
DOI: https://doi.org/10.1090/S0002-9947-99-02497-6
Published electronically: July 1, 1999
MathSciNet review: 1665338
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be an infinite $d$-regular graph and $L(G)$ its line graph. We consider discrete Laplacians on $G$ and $L(G)$, and show the exact relation between the spectrum of $-\Delta _G$ and that of $-\Delta _{L(G)}$. Our method is also applicable to $(d_1,d_2)$-semiregular graphs, subdivision graphs and para-line graphs.


References [Enhancements On Off] (What's this?)

  • 1. D. M. Cvetkovi\'{c}, M. Doob and H. Sachs. Spectra of graphs-theory and application, Deutscher Verlag der Wissenschaften, Berlin, and Academic Press, New York, 1980. MR 81i:05054
  • 2. J. Dodziuk and L. Karp. Spectral and function theory for combinatorial Laplacians, A. M. S. Contemporary Mathematics 73 (1988), 25-40. MR 89h:58220
  • 3. Yu. Higuchi. Isoperimetric inequality and random walks on an infinite graph and its line graph, preprint.
  • 4. Yu. Higuchi. Random Walks and Isoperimetric Inequalities on Infinite Planar Graphs and Their Duals, Dissertation, Univ. of Tokyo, January 1995.
  • 5. B. Mohar and W. Woess. A survey of spectra of infinite graphs, Bull. London Math. Soc. 21 (1989), 209-234. MR 90d:05162
  • 6. M. Reed and B. Simon. Methods of Modern Mathematical Physics, vol. I, Academic Press, New York, 1980. MR 85e:46002
  • 7. T. Shima. On eigenvalue problems for the random walks on the Sierpinski pre-gasket, Japan J. Indust. Appl. Math., 8 (1991), 127-141. MR 92g:60094

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Additional Information

Tomoyuki Shirai
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
Address at time of publication: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
Email: shirai@neptune.ap.titech.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-99-02497-6
Keywords: Regular line graph, subdivision, para-line graph, discrete Laplacian, spectrum
Received by editor(s): July 12, 1998
Published electronically: July 1, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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