Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-pcn4s Total loading time: 0.279 Render date: 2022-05-28T01:29:34.064Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

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
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, CanadaN2L 3G1

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
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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), 3960.CrossRefGoogle 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), 285289.CrossRefGoogle Scholar
[5] Godsil, C. D. and Gutman, I.. Note on bond orders. Z. Naturforschung, 39a (1984), 11841186.Google Scholar
[6] Godsil, C. D.. Spectra of trees. Annals Discrete Math., 20, (1984), 151159.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), 169, or Werke, III, pp. 191268.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. 153162.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. 849860.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. 1533.Google Scholar
[13] Yuan, Hong. An eigenvector condition for reconstructibility. J. Combinatorial Theory, Series B, 32 (1982), 353354.CrossRefGoogle Scholar
4
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

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

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *