Let $G$ be a graph of order $n$ with adjacency matrix $A(G)$. The eigenvalues of matrix $ S(G)=J_n-I_n-2A(G)$, where $J_n$ is the $n$ by $n$ matrix with all entries $1$, are called the Seidel eigenvalues of $G$. Let $\mathcal{G}(n,r)$ be the set of all graphs of order $n$ with a single Seidel eigenvalue with multiplicity $r$. In the present work, we will characterize all graphs in the class $\mathcal{G}(n,n-i)$ for $i=1,2$ and for the case $i=3$ our characterization is done by this condition that the nullity of $S(G)$ is zero. If the nullity of $S(G)$ is not zero the problem is solved in special cases.