bibkey: monterde2022strongcospectralitytwins
authors: Hermie Monterde
year: 2022
title: "Strong cospectrality and twin vertices in weighted graphs"
doi: 10.13001/ela.2022.6721
url: https://arxiv.org/abs/2111.01265v2
claim: "The paper studies strong cospectrality of vertices with respect to generalized adjacency and normalized adjacency matrices of weighted graphs, relates it to twin vertices, recalls the Arnadottir-Godsil bound |V(X)|/n on pairwise adjacency strongly cospectral vertices of normal Cayley graphs, where n is the largest eigenvalue multiplicity of A, and asks whether this bound holds for other classes of graphs and in general for generalized adjacency matrices."
strata_touched:
  - D5/S3/Quantum/Dynamics/StrongCospectralMultiplicityBoundRefutation
license: citation-only
triage: anchor

Strong cospectrality and twin vertices in weighted graphs

H. Monterde, arXiv:2111.01265v2; The Electronic Journal of Linear Algebra 38 (2022), 494–518. Quotations are from the v2 TeX source.

Generalized adjacency matrices:

Let $\alpha ,\beta ,\gamma \in \mathbb{R}$ with $\gamma \ne 0$. A generalized adjacency matrix $\text{A}(X)$ of $X$ is a matrix of the form $\text{A}(X)=\alpha I+\beta D(X)+\gamma A(X).$

Note that if $\alpha =\beta =0$ and $\gamma =1$, then $\text{A}=A$

Spectral decomposition and strong cospectrality:

Let $H$ be an $m\times m$ Hermitian matrix. Then $\sigma (H)\subseteq \mathbb{R}$ and $H$ admits a spectral decomposition $H={\sum }_j{\lambda }_j{E}_j$ where the ${\lambda }_j$’s are the distinct eigenvalues of $H$ and each $E_j$ is the orthogonal projection matrix onto the eigenspace associated with ${\lambda }_j$.

parallel if $E_j{\text{e}}_u$ and $E_j{\text{e}}_v$ are parallel vectors for each $j$, i.e., for each $j$, there exists $c\in \mathbb{C}$ such that $E_j{\text{e}}_u=c{E}_j{\text{e}}_v$,

strongly cospectral if for each $j$, (\ref{parallel}) holds for some $c\in \mathbb{C}$ with $|c|=1$.

If $H$ is real symmetric, then $u$ and $v$ are parallel with respect to $H$ if the constant $c$ in (\ref{parallel}) is real, and strongly cospectral with respect to $H$ if $E_j{\text{e}}_u=\pm E_j{\text{e}}_v$ for each $j$

The question (Section 3, PDF p. 15 of v2):

We mention an interesting result of Árnadóttir and Godsil which states that if $X$ is a normal Cayley graph and $n$ is the largest multiplicity of an eigenvalue of $A$, then the number of pairwise adjacency strongly cospectral vertices is bounded above by $|V(X)|/n$ [4, Theorem 6.1]. It would be interesting to check whether this bound holds for other classes of graphs, and in general whenever $M=\text{A}$.

Verified locator

Source: https://arxiv.org/abs/2111.01265v2

• URL: https://arxiv.org/abs/2111.01265v2 (the e-print is a tar archive; member ELA-sample1_copy.tex, md5 21a1ee53b9e64ae0b3a6773aa4fdbb45, 1444 lines):
  • generalized adjacency matrices, l. 143–150, and the adjacency special case, l. 161;
  • spectral decomposition, parallelism and strong cospectrality, l. 170–192;
  • the question after Theorem 3.11, l. 727.
• DOI: 10.13001/ela.2022.6721 (https://doi.org/10.13001/ela.2022.6721; Crossref lists The Electronic Journal of Linear Algebra, pages 494–518, published 10 September 2022, without a volume field; the arXiv journal reference gives volume 38).