Annihilating polynomial, Jordan canonical from, and generalized spectral characterizations of Eulerian graphs

Let $G$ be an Eulerian graph on $n$ vertices with adjacency matrix $A$ and characteristic polynomial $\phi (x)$. We show that when $n$ is even (resp. odd), the square-root of $\phi (x)$ (resp. $x\phi (x)$) is an annihilating polynomial of $A$, over ${\mathbb{F}}_2$. The result was achieved by applying the Jordan canonical form of $A$ over the algebraic closure ${\bar{\mathbb{F}}}_2$. Based on this, we show a family of Eulerian graphs are determined by their generalized spectrum among all Eulerian graphs, which significantly simplifies and strengthens the previous result.