The maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph

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.

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The maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph

Abstract: In this paper we introduce a parameter Mm(G), defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph G. We compute Mm(G) for several families of graphs.

Cited by 5 publications

References 15 publications

The liberation set in the inverse eigenvalue problem of a graph

The liberation set in the inverse eigenvalue problem of a graph

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Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

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