A Note on Non-$\mathbb{R}$-Cospectral Graphs

Two graphs $G$ and $H$ are called $\mathbb{R}$-cospectral if $A(G)+yJ$ and $A(H)+yJ$ (where $A(G)$, $A(H)$ are the adjacency matrices of $G$ and $H$, respectively, $J$ is the all-one matrix) have the same spectrum for all $y\in\mathbb{R}$. In this note, we give a necessary condition for having $\mathbb{R}$-cospectral graphs. Further, we provide a sufficient condition ensuring only irrational orthogonal similarity between certain cospectral graphs. Some concrete examples are also supplied to exemplify the main results.