In this paper we construct several classes of non-regular graphs which are co-spectral with respect to all the three matrices, namely, adjacency, Laplacian and normalized Laplacian, and hence we answer a question asked by S. Butler. We make these constructions starting with two pairs ($G_{1}$, $H_{1}$) and ($G_{2}$, $H_{2}$) of $A$-cospectral regular graphs, taking their $R$-graph $\mathcal{R}(G_{i})$, $\mathcal{R}(H_{i})$, $i=1,2$, and finally making some kind of partial joins between $\mathcal{R}(G_{1})$ and $\mathcal{R}(G_{2})$; and $\mathcal{R}(H_{1})$ and $\mathcal{R}(H_{2})$. Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs.