Minimum number of distinct eigenvalues of graphs

Abstract

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.

On Orthogonal Matrices with Zero Diagonal

“…It is not difficult to see that q(G) = 1 if and only if the graph G has no edges (see [1,Lemma 2.1]). So a natural question is to ask which graphs G have q(G) = 2.…”

“…So a natural question is to ask which graphs G have q(G) = 2. Various results on this question were obtained by Ahmadi et al in [1]: for instance, they showed that complete graphs K n , complete bipartite graphs K n,n and hypercubes Q n all have q(G) = 2. As observed in [1,Section 4], the property that q(G) = 2 for a non-null graph G is equivalent to the existence of an orthogonal matrix in S(G).…”

Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

“…The questions and studies mentioned above all fit under the general umbrella of the inverse eigenvalue problem (IEVP) of a graph, specifically looking at the achievable multiplicities of the eigenvalues in the IEVP. Many authors have provided partial answers to these questions; see [2,3,4,8,10,13]. Here, we add to the growing body of work, focusing on joins, complete multipartite graphs and the sets M P ([n − k, k]) for some k with k ≤ ⌊ n 2 ⌋.…”

“…If q(A) is the number of distinct eigenvalues of a symmetric matrix A, then for a given graph G, we let q(G) = min{q(A) : A ∈ S(G)}, and refer to this parameter as the minimum number of distinct eigenvalues of G. It is well-known that q(G) = 1 if and only if G has no edges, and q(G) = n if and only if G is a path on n vertices; see [2]. Graphs with q(G) = n − 1 are characterized in [4].…”

“…Theorem 4.4,[2]). Consider a connected graph G with q(G) = 2, and let S be an independent set of vertices.…”