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Let $R(H_1,H_2)$ denote the Ramsey number for the graphs $H_1, H_2$, and let$J_k$ be $K_k{-}e$. We present algorithms which enumerate all circulant andblock-circulant Ramsey graphs for different types of graphs, thereby obtainingseveral new lower bounds on Ramsey numbers including: $49 \leq R(K_3,J_{12})$,$36 \leq R(J_4,K_8)$, $43 \leq R(J_4,J_{10})$, $52 \leq R(K_4,J_8)$, $37 \leqR(J_5,J_6)$, $43 \leq R(J_5,K_6)$, $65\leq R(J_5,J_7)$. We also use a gluingstrategy to derive a new upper bound on $R(J_5,J_6)$. With both strategiescombined, we prove the value of two Ramsey numbers: $R(J_5,J_6)=37$ and$R(J_5,J_7)=65$. We also show that the 64-vertex extremal Ramsey graph for$R(J_5,J_7)$ is unique. Furthermore, our algorithms also allow to establish newlower bounds and exact values on Ramsey numbers involving wheel graphs andcomplete bipartite graphs, including: $R(W_7,W_4) = 21$, $R(W_7,W_7) = 19$,$R(K_{3,4},K_{3,4}) = 25$, and $R(K_{3,5}, K_{3,5})=33$.

New bounds for Ramsey numbers $R(K_k-e,K_l-e)$