The (normalized) Laplacian spectrum and related indexes of generalized quadrilateral graphs

Abstract

In this paper, we introduce the generalized quadrilateral graph Q⁽ⁿ⁾(G), which can be got by replacing each edge of the given graph G with a complete bipartite graph K_n,n. We characterize all the spectrum of the graph Q⁽ⁿ⁾(G) in terms of the given graph. Then we derive the formula for the multiplicative degree-Kirchhoff index, the Kemeny’s constant and the number of spanning trees of Q⁽ⁿ⁾(G). Finally, we can obtain more about the iterative graph Q_r⁽ⁿ⁾(G).