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Walk Generating Functions, Christoffel-Darboux Identities and the Adjacency Matrix of a Graph

Published online by Cambridge University Press:  12 September 2008

C. D. Godsil
C. D. Godsil
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, CanadaN2L 3G1
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Abstract

In this work we show that that many of the basic results concerning the theory of the characteristic polynomial of a graph can be derived as easy consequences of a determinantal identity due to Jacobi. As well as improving known results, we are also able to derive a number of new ones. A combinatorial interpretation of the Christoffel-Darboux identity from the theory of orthogonal polynomials is also presented. Finally, we extend some work of Tutte on the reconstructibility of graphs with irreducible characteristic polynomials.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 1 , March 1992 , pp. 13 - 25
Copyright
Copyright © Cambridge University Press 1992

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References

[1] Aitken, A. C.. Determinants and Matrices, 9th ed.Oliver and Boyd, Edinburgh (1956).Google Scholar
[2] Coulson, C. A. and Longuet-Higgins, H. C.. The electronic structure of conjugated systems I. General theory. Proc. Roy. Soc. London A191 (1947), 39–60.Google Scholar
[3] Cvetković, D., Doob, D. M. and Sachs, H.. Spectra of Graphs. Academic Press, New York (1980).Google Scholar
[4] Godsil, C. D. and McKay, B. D.. Spectral conditions for the reconstructibility of a graph. J. Combinatorial Theory B, 30 (1981), 285–289.CrossRefGoogle Scholar
[5] Godsil, C. D. and Gutman, I.. Note on bond orders. Z. Naturforschung, 39a (1984), 1184–1186.CrossRefGoogle Scholar
[6] Godsil, C. D.. Spectra of trees. Annals Discrete Math., 20, (1984), 151–159.Google Scholar
[7] Jacobi, C. G. J.. De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; una cum variis theorematis de transformatione et determinatione integralium multiplicium. Crelle's J., 12 (1833), 1–69, or Werke, III, pp. 191–268.Google Scholar
[8] Schwenk, A. J.. Computing the characteristic polynomial of a graph. In Graphs and Combinatorics. Lecture Notes in Mathematics 406, (Springer Verlag, Berlin) 1974, pp. 153–162.CrossRefGoogle Scholar
[9] Schwenk, A. J.. Removal cospectral sets of points in a graph. In Proc. 10th S-E Conf. Combinatorics, Graph Theory and Computing, pp. 849–860.Google Scholar
[10] Schwenk, A. J.. The adjoint of the adjacency matrix of a graph. Preprint (1987).Google Scholar
[11] Szegö, G.. Orthogonal Polynomials. American Math. Society, Providence (1975).Google Scholar
[12] Tutte, W. T.. All the king's horses. In Graph Theory and Related Topics, edited by Bondy, J. A. and Murty, U. S. R.. Academic Press, New York (1979), pp. 15–33.Google Scholar
[13] Yuan, Hong. An eigenvector condition for reconstructibility. J. Combinatorial Theory, Series B, 32 (1982), 353–354.CrossRefGoogle Scholar