Primary decomposition theorem and generalized spectral characterization of graphs
Songlin Guo, Wei Wang, Wei Wang
TL;DR
This paper addresses when a controllable graph $G$ is determined by its generalized spectrum (DGS). It builds a new criterion that removes the previous squarefreeness requirement on $\theta(G)=\gcd\{2^{-\lfloor n/2\rfloor}\det W,\Delta\}$ by leveraging the Primary Decomposition Theorem over $\mathbb{F}_p$, together with the structure of rational regular orthogonal transformations. The main result shows that for each odd prime factor $p$ of $\theta(G)$ and each multiple irreducible factor of $\Phi_p(G;x)$, a certain inequality involving $\det\phi(A)$ and $\det\phi(A+J)$ must hold; if so, $G$ is DGS. This extends prior sufficient conditions (Exclusion Condition, Improved Condition) and broadens the class of graphs identifiable by generalized spectral data, as demonstrated by several examples and computational experiments. The work advances the toolbox for graph isomorphism via spectral data by integrating primary decomposition-based arguments with finite-field invariants.
Abstract
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.