The Laplacian polynomial and Kirchhoff index of graphs derived from regular graphs

Let R(G)R(G) be the graph obtained from GG by adding a new vertex corresponding to each edge of GG and by joining each new vertex to the end vertices of the corresponding edge, and Q(G)Q(G) be the graph obtained from GG by inserting a new vertex into every edge of GG and by joining by edges those pairs of these new vertices which lie on adjacent edges of GG. In this paper, we determine the Laplacian polynomials of R(G)R(G) and Q(G)Q(G) of a regular graph GG; on the other hand, we derive formulae and lower bounds of the Kirchhoff index of these graphs.