In this paper, we determine the bounds about Ramsey number $R(W_m, W_n),$ where $W_i$ is a graph obtained from a cycle $C_i$ and an additional vertex by
joining it to every vertex of the cycle $C_i.$ We prove that $3m+1 ≤ R(W_m, W_n) ≤
8m − 3$ for odd $n,$ $m ≥ n ≥ 3,$ $m ≥ 5,$ and $2m + 1 ≤ R(W_m, W_n) ≤ 7m − 2$ for
even $n$ and $m ≥ n + 502.$ Especially, if $m$ is sufficiently large and $n = 3,$ we
have $R(W_m, W_3) = 3m + 1.$
