On the normalized Laplacians with some classical parameters involving graph transformations

Given a connected graph G, two types of graph transformations on G are considered. The graph is obtained by applying the first transformation on G, i.e. it is formed by adding a new triangle for each edge e=uv in G and then adding in edges and , whereas the graph is obtained by applying the second transformation on G, i.e. it is formed by adding a new quadrangle for each edge e=uv in G and then adding in edges and . Repeating the above constructions r times yields the iterative graphs and . In this paper, the normalized Laplacian spectrum of (resp. ) is completely determined in regards to G. As applications, the significant formula are obtained to calculate the multiplicative degree-Kirchhoff index, the Kemeny's constant and the number of spanning trees of the rth iterative graph (resp. ) compared to those of G, where .