Star sets and related aspects of algebraic graph theory - PhDData

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## Star sets and related aspects of algebraic graph theory

The thesis was published by Jackson, Penelope S, in September 2022, University of Stirling.

Abstract:

Let ÎĽ be an eigenvalue of the graph G with multiplicity k. A star set corresponding to ÎĽ in G is a subset of V(G) such that [x] = k and ÎĽ 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 ÎĽ Îµ {-I, a}.

The induced subgraph H = G – X is called a star complement for ÎĽ in G. The Reconstruction Theorem states that for a given eigenvalue ÎĽ of G, knowledge of a star complement corresponding to ÎĽ, 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 ÎĽ is constrained by the condition that ÎĽ 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.

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