Please use this identifier to cite or link to this item: http://hdl.handle.net/1893/26690
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dc.contributor.authorJackson, Penelope S-
dc.date.accessioned2018-02-13T16:41:45Z-
dc.date.available2018-02-13T16:41:45Z-
dc.date.issued1999-
dc.identifier.urihttp://hdl.handle.net/1893/26690-
dc.description.abstractLet 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.en_GB
dc.language.isoenen_GB
dc.publisherUniversity of Stirlingen_GB
dc.subject.lcshEigenvaluesen_GB
dc.subject.lcshGraph theoryen_GB
dc.titleStar sets and related aspects of algebraic graph theoryen_GB
dc.typeThesis or Dissertationen_GB
dc.type.qualificationlevelDoctoralen_GB
dc.type.qualificationnameDoctor of Philosophyen_GB
Appears in Collections:Computing Science and Mathematics eTheses

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