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Mathematics 2019, 7(4), 314; https://doi.org/10.3390/math7040314

On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

1
School of Mathematical Sciences, Anhui Jianzhu University, Hefei 230601, China
2
School of Mathematics, Southeast University, Nanjing 210096, China
3
Department of Mathematics and Statistics, South Central University for Nationalities, Wuhan 430074, China
*
Author to whom correspondence should be addressed.
Received: 22 February 2019 / Revised: 24 March 2019 / Accepted: 25 March 2019 / Published: 28 March 2019
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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PDF [373 KB, uploaded 28 March 2019]
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Abstract

The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let H n be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of H n due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of H n by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to H n . View Full-Text
Keywords: normalized Laplacian; resistance distance; degree-Kirchhoff index; spanning tree normalized Laplacian; resistance distance; degree-Kirchhoff index; spanning tree
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Liu, J.-B.; Zhao, J.; Zhu, Z.; Cao, J. On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks. Mathematics 2019, 7, 314.

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