On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results

Abstract

Simic, Slobodan K.; Andelic, Milica; Da Fonseca, Carlos M.; and Zivkovic, Dejan. (2015) Abstract. Given a simple graph G, let A G be its adjacency matrix. A principal submatrix of A G of order one less than the order of G is the adjacency matrix of its vertex deleted subgraph. It is well-known that the multiplicity of any eigenvalue of A G and such a principal submatrix can differ by at most one. Therefore, a vertex v of G is a downer vertex (neutral vertex, or Parter vertex) with respect to a fixed eigenvalue µ if the multiplicity of µ in A G−v goes down by one (resp., remains the same, or goes up by one). In this paper, we consider the problems of characterizing these three types of vertices under various constraints imposed on graphs being considered, on vertices being chosen and on eigenvalues being observed. By assigning weights to edges of graphs, we generalize our results to weighted graphs, or equivalently to symmetric matrices.

On the spectral invariants of symmetric matrices with applications in the spectral graph theory

“…Using the generalized form of Rowlinson's formula (given in [11], see also [6,9] for its original form) we obtain…”

Coalescing Fiedler and core vertices

“…There are three types of vertices, depending on the change in the multiplicity of an eigenvalue allowed by the Interlacing theorem. In [11], [23], a vertex is referred to as downer, neutral or Parter when its deletion decreases, does not change or increases the multiplicity of an eigenvalue, respectively. For the eigenvalue zero, Fiedler vertices and Parter vertices are sometimes referred to as F-vertices and P-vertices, respectively, see [1].…”

Graphs with least eigenvalue −2: Ten years on

“…k, k + 1). For μ = −2, the vertices of line graphs are classified accordingly in [101]. The occurrence of repeated eigenvalues in the line graph of a tree is considered in [100], in the context of the polynomial reconstruction problem [4,Section 7.5].…”