Let G be a graph. The edge blow-up graph \(G^{p+1}\) of G is a graph obtained by replacing each edge of G with a clique of order \(p+1\) , where the new vertices of the cliques are all distinct. Denote by \(K_t^-\) the graph obtained from the complete graph \(K_t\) by removing one edge. In this paper, we construct two classes of edge blow-up graphs, namely the string graph and the ring graph associated with \(K_t^-\) , which are denoted \(S(K_r^-, K_m^-, n)\) and \(R(K_r^-, K_m^-, n)\) , respectively, with \(r, m \ge 4\) . Specifically, for a path \(P_{n}\) , we alternately replace each edge of \(P_{n}\) with \(K_r^-\) and \(K_m^-\) . For a cycle \(C_{n}\) , we alternately replace each edge of \(C_{n}\) with \(K_r^-\) and \(K_m^-\) , where n is even. This construction produces the string graph \(S(K_r^-, K_m^-, n)\) and the ring graph \(R(K_r^-, K_m^-, n)\) . Using combinatorial and electrical network methods, we derive formulas for the resistance distances in these two classes of graphs.
