Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

Abstract

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.

Orthogonal symmetric matrices and joins of graphs

“…It is straightforward to see that 2K 2 and 3K 1 do not have compatible multiplicity matrices, hence q(2K 2 ∨ 3K 1 ) ≠ 2. This gives an explicit counterexample to the claim in [1,Lemma 3.4], which was later retracted in [2]. In contrast, we will see in Section 4 that q(2K 2 ∨ 2K 1 ) = q(2K 2 ∨ 4K 1 ) = 2.…”

“…The same authors also proved that q(G ∨ G) = 2 for any connected graph G. Later, this result was generalized by Monfared and Shader [18], who proved that q(G ∨ H) = 2 for any connected graphs G and H with the same number of vertices. Recently, joins of disconnected graphs were investigated in [1,2], where particular attention was given to joins of unions of complete graphs.…”

Paths are generically realisable

“…Let M(t) ∶= M f,G (t) be the smooth family of manifolds of m × m symmetric matrices defined in (1). Note that M(0) is the set of diagonal matrices.…”

“…For some specific families of connected graphs, several ordered multiplicity vectors have been determined (see e.g. [1,5,18]). Moreover, Monfared and Shader proved the following theorem in [19], showing that (1, 1, .…”

Tight Frame Graphs Arising as Line Graphs

“…It remains an open question (see [2]) whether the implication holds for graphs Γ with msr(Γ) > 2. In particular, since |L (K n )| = 1 2 n(n − 1), it remains to be determined whether L (K n ) is a tight frame graph for H d for n ≤ d ≤ ⌊ 1 4 n(n − 1)⌋.…”