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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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