Isomorphisms between Jacobson graphs

Let $R$ be a commutative ring with a non-zero identity and $\mathfrak{J}_R$ be its Jacobson graph. We show that if $R$ and $R'$ are finite commutative rings, then $\mathfrak{J}_R\cong\mathfrak{J}_{R'}$ if and only if $|J(R)|=|J(R')|$ and $R/J(R)\cong R'/J(R')$. Also, for a Jacobson graph $\mathfrak{J}_R$, we obtain the structure of group $\mathrm{Aut}(\mathfrak{J}_R)$ of all automorphisms of $\mathfrak{J}_R$ and prove that under some conditions two semi-simple rings $R$ and $R'$ are isomorphic if and only if $\mathrm{Aut}(\mathfrak{J}_R)\cong\mathrm{Aut}(\mathfrak{J}_{R'})$.

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