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?G2 G 1 ? G 2 , is the graph obtained from vertex disjoint R(G1) R ( G 1 ) and |V(G1)| | V ( G 1 ) | copies of G2 by joining the ith vertex of V(G1) V ( G 1 ) to every vertex in the ith copy of G2. The R-edge corona of G1 and G2, denoted by G1?G2 G 1 ? G 2 , is the graph obtained from vertex disjoint R(G1) R ( G 1 ) and |I(G1)| | I ( G 1 ) | copies of G2 by joining the ith vertex of I(G1) I ( G 1 ) 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?G2 G 1 ? G 2 and G1?G2 G 1 ? G 2 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.
laplacian polynomial kirchhoff index corona resistance distance edge corona schur complement