The combinatorial inverse eigenvalue problems: complete graphs and small graphs with strict inequality

Abstract

Abstract. Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n × n matrices whose nonzero off-diagonal entries correspond exactly to the edges of G.and a vertex v of G, the question is addressed of whether or not there exists A ∈ S(G) with eigenvalues λ 1 , . . . , λn such that A(v) has eigenvalues µ 1 , . . . , µ n−1 , where A(v) denotes the matrix with the vth row and column deleted. General results that apply to all connected graphs G are given first, followed by a complete answer to the question for Kn. Since the answer is constructive it can be implemented as an algorithm; a Mathematica code is provided to do so. Finally, for all connected graphs on 4 vertices it is shown that the answer is affirmative if all six inequalities are strict.

Multiplicity lists for symmetric matrices whose graphs have few missing edges

“…This fact has been known for some time, having been noticed by the author Johnson and mentioned in talks by him on the subject for many years. The same has also been noticed much later in [2]. Here we consider it as a starting point.…”

“…This is so, and we will discuss a few natural cases here, e.g., the complete graph missing just one or two edges. The next case, the complete graph, less one edge was left as an open question in [2].…”

On the inverse eigenvalue problem for block graphs

“…Choosing B ∈ S(K k ) in Lemma 2.1 results in k-duplication in the associated graph. While this lemma can be applied more generally, we will take particular advantage of the fact that the IEP-G for complete graphs is solved (see e.g., [5]). Furthermore, we will need information on possible patterns of the eigenvectors of matrices in S(K k ), as outlined in the following Lemma.…”

“…The solutions of the IEP-G for generalized star graphs [17] and cycles [8] are also known. The IEP-G for complete graphs and small graphs (up to 4 vertices) was solved by explicit matrix construction in [5,6]. Recently, Barrett et al introduced new techniques to the problem based on the strong spectral property (SSP), and solved the IEP-G for graphs with up to 5 vertices [3].…”

Generalizations of the Strong Arnold Property and the Minimum Number of Distinct Eigenvalues of a Graph

“…The inverse spectrum problem for G seems to be difficult, as evidenced by that fact that it has been completely solved for only a few special families of graph, e.g. paths, generalized stars, double generalized stars, and complete graphs [7,9,21].…”