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http://hdl.handle.net/1893/26690
Appears in Collections: | Computing Science and Mathematics eTheses |
Title: | Star sets and related aspects of algebraic graph theory |
Authors: | Jackson, Penelope S |
Issue Date: | 1999 |
Publisher: | University of Stirling |
Abstract: | Let μ be an eigenvalue of the graph G with multiplicity k. A star set corresponding to μ in G is a subset of V(G) such that [x] = k and μ is not an eigenvalue of G - X. It is always the case that the vertex set of G can be partitioned into star sets corresponding to the distinct eigenvalues of G. Such a partition is called a star partition. We give some examples of star partitions and investigate the dominating properties of the set V (G) \ X when μ ε {-I, a}. The induced subgraph H = G - X is called a star complement for μ in G. The Reconstruction Theorem states that for a given eigenvalue μ of G, knowledge of a star complement corresponding to μ, together with knowledge of the edge set between X and its complement X, is sufficient to reconstruct G. Pursuant to this we explore the idea that the adjacencies of pairs of vertices in X is determined by the relationship between the H-neighbourhoods of these vertices. We give some new examples of cubic graphs in this context. For a given star complement H the range of possible values for the corresponding eigenvalue μ is constrained by the condition that μ must be a simple eigenvalue of some one-vertex extension of H, and a double eigenvalue of some two-vertex extension of H. We apply the Reconstruction Theorem to the generic form of a two-vertex extension of H, thereby obtaining sufficient information to construct a graph containing H as a star complement for one of the possible eigenvalues. We give examples of graph characterizations arising in the case where the star complement is (to within isolated vertices) a complete bipartite graph. |
Type: | Thesis or Dissertation |
URI: | http://hdl.handle.net/1893/26690 |
Files in This Item:
File | Description | Size | Format | |
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Jackson-thesis.pdf | 4.2 MB | Adobe PDF | View/Open |
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