Primary decomposition theorem and generalized spectral characterization of graphs

Abstract

Suppose $G$ is a controllable graph of order $n$ with adjacency matrix $A$. Let $W=[e,Ae,\dots ,A^{n-1}e]$ ($e$ is the all-one vector) and $\hat{I}”={\prod }_{i>j}(\hat{I}{\pm }_i-\hat{I}{\pm }_j{)}^2$ ($\hat{I}{\pm }_i$'s are eigenvalues of $A$) be the walk matrix and the discriminant of $G$, respectively. Wang and Yu \cite{wangyu2016} showed that if $$\hat{I}¸(G):=\gcd \{2^{-\lfloor \frac{n}{2}\rfloor }\det W,\hat{I}”\}$$ 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 $\hat{I}¸(G)$. 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.