On the Laplacian, the Kirchhoff Index, and the Number of Spanning Trees of the Linear Pentagonal Derivation Chain

Let Pn be a pentagonal chain with 2n pentagons in which two edges common can regarded as adding one vertex and to hexagon. Thus, the linear derivation chains QPn represent graph obtained by attaching four-membered rings every of Pn. In this article, Laplacian spectrum consisting eigenvalues symmetric matrices is determined. Next, formulas for invariants that represented spectrum, namely, Kirchhoff index number spanning trees, are studied. Surprisingly, almost half Wiener QPn.