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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 2372931, 10 pages
Research Article

Generalized Characteristic Polynomials of Join Graphs and Their Applications

School of Computer and Communication, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Correspondence should be addressed to Pengli Lu; moc.361@88ilgnepul

Received 11 November 2016; Revised 18 January 2017; Accepted 26 January 2017; Published 2 March 2017

Academic Editor: Francisco R. Villatoro

Copyright © 2017 Pengli Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Kirchhoff index of is the sum of resistance distances between all pairs of vertices of in electrical networks. is the Laplacian-Energy-Like Invariant of in chemistry. In this paper, we define two classes of join graphs: the subdivision-vertex-vertex join and the subdivision-edge-edge join . We determine the generalized characteristic polynomial of them. We deduce the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomials of and when is -regular graph and is -regular graph. As applications, the Laplacian spectra enable us to get the formulas of the number of spanning trees, Kirchhoff index, and of and in terms of the Laplacian spectra of and .