Abstract: Given two graphs G and H, the Ramsey number R(G,H) is the smallest positive integer N such that every 2-coloring of the edges of K_N contains either a red G or a blue H. Let K_N-1\sqcup K_1,\,k be the graph obtained from K_N-1 by adding a new vertex v connecting k vertices of K_N-1. A graph G with \chi(G)=k+1 is called edge-critical if G contains an edge e such that \chi(G-e)=k. A considerable amount of research has been conducted by previous scholars on Ramsey numbers of graphs. In this study, we show that for an edge-critical graph G with \chi(G)=k+1, when k\geq 2, t\geq 2, and n is sufficiently large, R(G, K_1+nK_t)=knt+1 and r_*(G,K_1+nK_t)=(k-1)nt+t.