Some consequences of an inequality on the spectral multiplicity of graphs

Abstract

We present two distinct applications of an inequality relating the multiplicity of an eigenvalue of a graph to a certain subgraph. The first is related to a recent classification, established by Kim and Shader, for the class of those trees for which each of the associated matrices have distinct eigenvalues whenever the diagonal entries are distinct. We analyze the minimum number of distinct diagonal entries and the corresponding location, in order to preserve such multiplicity characterization. The second application involves a new property of a star set of a graph due to P. Rowlinson.

Unordered multiplicity lists of wide double paths

“…Finally, a tree is minimal provided there is a matrix such that number of distinct eigenvalues is equal to the diameter (counting edges) plus one. From Theorem 4.1, we conclude that the wide double paths are minimal, for 2 , 4 1 , 3 5 . In fact, with t 3 = 0, t 2 = 2 + 4 , and t 1 = ( 1 − 2 ) + ( 3 − 4 ) + 5 + 2, since the number of distinct eigenvalues is 1 + 3 + 5 + 2 and the diameter is 1 + 3 + 5 + 1.…”

“…We remark that the Parter-Wiener Theorem was reformulated in the survey work [7], by the second author, motivated by the earlier seminal work of C. Godsil on matchings polynomials [9,10,11]. The same approach produced a result for the multiplicities of an eigenvalue of a matrix involving certain paths of the underlying graph, with many interesting applications to general graphs [4]. Theorem 1.1.…”

Graphs with least eigenvalue −2: Ten years on

“…Here we note that star complements have been used to establish sharp upper bounds for eigenvalue multiplicity in further classes of graphs, including trees [88], cubic graphs [95], and graphs with prescribed girth [89]. Some results from [88] have been refined in [46].…”