Star sets and related aspects of algebraic graph theory

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

On some algorithmic investigations of star partitions of graphs

AbstractStar partitions of graphs were introduced in a recent paper by the same authors in order to extend spectral methods in algebraic graph theory. Here it is shown that the corresponding partitioning problem is polynomial. Two algorithms are investigated: the first is based on the maximum matching problem for graphs, and the second invokes an algorithm for matroid intersection