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-ones vector) and $\Delta ={\prod }_{i>j}{({\alpha }_i-{\alpha }_j)}^2$ (${\alpha }_i$'s are eigenvalues of A) be the walk matrix and the discriminant of G, respectively. Wang and Yu (arXiv:1608.01144) [21] showed that if$\theta \left(G\right):=\gcd \{2^{-\lfloor \frac{n}{2}\rfloor }\det W,\Delta \}$ 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 $\theta \left(G\right)$. 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.