The Laplacian polynomial and Kirchhoff index of graphs based on R-graphs

Let R(G)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 I(G)I(G) be the set of newly added vertices. The R-vertex corona of G1 and G2, denoted by G1⊙G2G1⊙G2, is the graph obtained from vertex disjoint R(G1)R(G1) and |V(G1)||V(G1)| copies of G2 by joining the i  th vertex of V(G1)V(G1) to every vertex in the ith copy of G2. The R-edge corona of G1 and G2, denoted by G1⊖G2G1⊖G2, is the graph obtained from vertex disjoint R(G1)R(G1) and |I(G1)||I(G1)| copies of G2 by joining the i  th vertex of I(G1)I(G1) to every vertex in the ith copy of G2. Liu et al. gave formulae for the Laplacian polynomial and Kirchhoff index of RT(G)RT(G) in [19]. In this paper, we give the Laplacian polynomials of G1⊙G2G1⊙G2 and G1⊖G2G1⊖G2 for a regular graph G1 and an arbitrary graph G2; on the other hand, we derive formulae and lower bounds of Kirchhoff index of these graphs and generalize the existing results.