Resistance distance and Kirchhoff index of two kinds of double join operations on graphs

Let $G$ be a connected graph. The resistance distance between any two vertices of $G$ is defined to be the network effective resistance between them if each edge of $G$ is replaced by a unit resistor. The Kirchhoff index of $G$ is the sum of resistance distances between all pairs of vertices of $G$. In this paper, we determine the resistance distance and Kirchhoff index of the subdivision double join $G^{S}\vee\{G_{1},G_{2}\}$ and $R$-graph double join $G^{R}\vee\{G_{1},G_{2}\}$ for a regular graph $G$ and two arbitrary graphs $G_1$, $G_2$, respectively.