  Let $R(G)$ be the graph obtained from $G$ by adding a new vertex
corresponding to each edge of $G$ and by joining each new vertex to the end
vertices of the corresponding edge. Let $RT(G)$ be the graph obtained from
$R(G)$ by adding a new edge corresponding to every vertex of $G$, and by
joining each new edge to every vertex of $G$. In this paper, we determine the
Laplacian polynomials of $RT(G)$ of a regular graph $G$. Moreover, we derive
formulae and lower bounds of Kirchhoff index of the graphs. Finally we also
present the formulae for calculating the Kirchhoff index of some special graphs
as applications, which show the correction and efficiency of the proposed
results.
