Estimation of the maximum multiplicity of an eigenvalue in terms of the vertex degrees of the graph of a matrix

Abstract

The maximum multiplicity among eigenvalues of matrices with a given graph cannot generally be expressed in terms of the degrees of the vertices (even when the graph is a tree). Given are best possible lower and upper bounds, and characterization of the cases of equality in these bounds. A by-product is a sequential algorithm to calculate the exact maximum multiplicity by simple counting.

On the multiplicities of eigenvalues of a Hermitian matrix whose graph is a tree

“…Johnson and Saiago [17] gave tight bounds for ∆(T ) in terms of all vertex whose degrees are greater than 2. This result also comes out as a corollary of the Lemma 2.2.…”

“…Johnson, A. Leal Duarte and others (cf. [13,14,15,16,17,18]) inspired by the work of J. Genin and J.S. Maybee [6] and S. Parter [21].…”

“…One of the concepts explored by Johnson, Leal Duarte and Saiago [13,17] is the minimum number of vertex disjoint paths, occurring as induced subgraphs of a tree T , which cover all the vertices of T . This concept had already been studied by O.…”

Minimum rank of a graph over an arbitrary field

“…This problem has received considerable attention recently; see [1], [2], [3], [4], [5], [7], [9], [10], [11], [13], [14], [15]. Originally the minimum rank problem was studied over the real numbers, where it is equivalent to the question of maximum multiplicity of an eigenvalue of the same family of matrices.…”

“…Define the minimum rank of G over F as In 1996 Nylen gave a method for computation of minimum rank for a tree, subsequently improved by Johnson and Leal-Duarte [11], Johnson and Saiago [14], and others. Convenient algorithms for computation of the minimum rank (over the reals) of a tree (by computation of the graph parameter ∆, cf.…”