On the Laplacian, the Kirchhoff Index, and the Number of Spanning Trees of the Linear Pentagonal Derivation Chain
Author:
Tu Yue,Ma Xiaoling,Zhang Yuqing,Ren Junyu
Abstract
Let Pn be a pentagonal chain with 2n pentagons in which two pentagons with two edges in common can be regarded as adding one vertex and two edges to a hexagon. Thus, the linear pentagonal derivation chains QPn represent the graph obtained by attaching four-membered rings to every two pentagons of Pn. In this article, the Laplacian spectrum of QPn consisting of the eigenvalues of two symmetric matrices is determined. Next, the formulas for two graph invariants that can be represented by the Laplacian spectrum, namely, the Kirchhoff index and the number of spanning trees, are studied. Surprisingly, the Kirchhoff index is almost one half of the Wiener index of a linear pentagonal derivation chain QPn.
Funder
the Natural Science Foundation of Xinjiang Province
Subject
Geometry and Topology,Logic,Mathematical Physics,Algebra and Number Theory,Analysis
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