Aspects of Graph Spectra  

Abstract Category: Science
Course / Degree: Ph.D. (Mathematics)
Institution / University: University of Malta, Malta
Published in: 1999

Thesis Abstract / Summary:

The main themes of this thesis are
i. characterisations of singular graphs,
ii. the study of the spectrum of the line graphs of trees, and
iii. the determination of the characteristic polynomial of certain classes of graphs. Aspects of the polynomial reconstruction theme appear throughout.

In the first part, we define a core, chi, of a singular graph, corresponding to a kernel eigenvector x to be the subgraph induced by the vertices corresponding to the non-zero components of x. Every singular graph contains a minimal configuration Gamma, `grown' from some core chi, by successively adding vertices until the nullity of Gamma is reduced to one. We characterise singular graphs in terms of the minimal configurations. Their determination has various applications, particularly in chemistry.

The nut graphs are the minimal conßgurations whose core spans the whole graph. We show that nut graphs exist for all orders n >= 7. They prove to be crucial in the classification of core-spaces and in the study of the rank of graphs.

In the second part, we present an algorithm, which analyses the p-deck, PD(H), of characteristic polynomials of the one-vertex-deleted subgraphs of a graph H, and which not only recognises and reconstructs the line graph, H = LG, of a triangle-free graph, G, but also extracts the root graph G. We then show that the geometry of the line graphs of trees, determines the multiplicity of certain eigenvalues including zero, the golden section, and  1. Here also, nut graphs play an interesting role.

In the third part, the determination of the characteristic polynomial, phi(G); of a graph G is considered. We supplement PD(G) with other parameters of G to determine if G is disconnected and thus form phi(G). Finally we express the characteristic polynomial of a homeomorph H(S) in terms of those of S and some of its subgraphs and apply the results to improve on the Chartrand-Harary Theorem which characterises outerplanar graphs in terms of forbidden subgraphs.

Thesis Keywords/Search Tags:
Eigenvalues, Graph Spectra, Singular Graphs, Minimal Configurations

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Submission Details: Thesis Abstract submitted by Irene Sciriha from Malta on 04-Nov-2003 08:06.
Abstract has been viewed 2688 times (since 7 Mar 2010).

Irene Sciriha Contact Details: Email: isci1@um.edu.mt

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