The normalized Laplacian spectrum of quadrilateral graphs and its applications

Abstract The quadrilateral graph Q ( G ) of G is obtained from G by replacing each edge in G with two parallel paths of lengths 1 and 3. In this paper, we completely describe the normalized Laplacian spectrum on Q ( G ) for any graph G . As applications, the significant formulae to calculate the multiplicative degree-Kirchhoff index, the Kemeny’s constant and the number of spanning trees of Q ( G ) and the quadrilateral iterative graph Q r ( G ) are derived.

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