Abstract. The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to deduce further properties of q(G). It is shown that there is a great number of graphs G for which q(G) = 2. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs G are provided to show that adding and deleting edges or vertices can dramatically change the value of q(G). Finally, the set of graphs G with q(G) near the number of vertices is shown to be a subset of known families of graphs with small maximum multiplicity.

We survey results concerning reconfigurations of colourings and dominating sets in graphs. The vertices of the k-colouring graph C k (G) of a graph G correspond to the proper k-colourings of a graph G, with two k-colourings being adjacent whenever they differ in the colour of exactly one vertex. Similarly, the vertices of the k-edge-colouring graph EC k (G) of g are the proper k-edge-colourings of G, where two k-edge-colourings are adjacent if one can be obtained from the other by switching two colours along an edge-Kempe chain, i.e., a maximal two-coloured alternating path or cycle of edges.The vertices of the k-dominating graph D k (G) are the (not necessarily minimal) dominating sets of G of cardinality k or less, two dominating sets being adjacent in D k (G) if one can be obtained from the other by adding or deleting one vertex. On the other hand, when we restrict the dominating sets to be minimum dominating sets, for example, we obtain different types of domination reconfiguration graphs, depending on whether vertices are exchanged along edges or not.We consider these and related types of colouring and domination reconfiguration graphs. Conjectures, questions and open problems are stated within the relevant sections. * Supported by the Natural Sciences and Engineering Research Council of Canada.1 We use the term connectedness instead of connectivity when referring to the question of whether a graph is connected or not, as the latter term refers to a specific graph parameter.

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Associated to a graph G is a set S(G) of all real-valued symmetric matrices whose off-diagonal entries are non-zero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in S(G) partition n; this is called a multiplicity partition.We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in S(G) with partitions [n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in S(G) with multiplicity partition [n − k, k] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in S(G) with multiplicity partition [n − k, k] to show the complexities of characterizing these graphs.