Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.12932 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Suppose $G$ is a controllable graph of order $n$ with adjacency matrix $A$. Let $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector) and $Δ=\prod_{i>j}(α_i-α_j)^2$ ($α_i$'s are eigenvalues of $A$) be the walk matrix and the discriminant of $G$, respectively. Wang and Yu \cite{wangyu2016} showed that if $$θ(G):=\gcd\{2^{-\lfloor\frac{n}{2}\rfloor}\det W,Δ\} $$ is odd and squarefree, then $G$ is determined by its generalized spectrum (DGS). Using the primary decomposition theorem, we obtain a new criterion for a graph $G$ to be DGS without the squarefreeness assumption on $θ(G)$. Examples are further given to illustrate the effectiveness of the proposed criterion, compared with the two existing methods to deal with the difficulty of non-squarefreeness.