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

  • C. D. Godsil (a1)
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.

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[2] C. A. Coulson and H. C. Longuet-Higgins . The electronic structure of conjugated systems I. General theory. Proc. Roy. Soc. London A191 (1947), 3960.

[4] C. D. Godsil and B. D. McKay . Spectral conditions for the reconstructibility of a graph. J. Combinatorial Theory B, 30 (1981), 285289.

[8] A. J. Schwenk . Computing the characteristic polynomial of a graph. In Graphs and Combinatorics. Lecture Notes in Mathematics 406, (Springer Verlag, Berlin) 1974, pp. 153162.

[13] Hong Yuan . An eigenvector condition for reconstructibility. J. Combinatorial Theory, Series B, 32 (1982), 353354.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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