Resistance Distance and Kirchhoff Index of $Q$ -Double Join Graphs

Let Q(G) be the graph derived from G by inserting a new vertex into every edge of G and by connecting edges of these new vertices that are on adjacent edges of G, and Q-double join of G, G
₁
and G
₂
denoted by G
^Q
v{G
₁
, G
₂
}, is the graph derived from Q(G), G
₁
and G
₂
, by connecting every old-vertex V (G) of Q(G) with every vertex of G
₁
and every new-vertex I(G) of Q(G) with every vertex of G
₂
. In this paper, we mainly considered the resistance distance and Kirchhoff index of Q-double join graphs G
^Q
v {G
₁
, G
₂
} of an r-regular graph G and any two graphs of G
₁
and G
₂
.