Generalized Characteristic Polynomials of Join Graphs and Their Applications

Generalized Characteristic Polynomials of Join Graphs and Their Applications

The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices of G in electrical networks. LEL(G) is the Laplacian-Energy-Like Invariant of G in chemistry. In this paper, we define two classes of join graphs: the subdivision-vertex-vertex join G1⊚G2 and the subdivision-ed...

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Main Authors: Pengli Lu, Ke Gao, Yang Yang
Format: Article
Language:English
Published: Hindawi Limited 2017-01-01
Series:Discrete Dynamics in Nature and Society
Online Access:http://dx.doi.org/10.1155/2017/2372931
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spelling doaj-ce381ae1e6e54c0ebedc2f3e795f4ecd2020-11-24T22:23:38ZengHindawi LimitedDiscrete Dynamics in Nature and Society1026-02261607-887X2017-01-01201710.1155/2017/23729312372931Generalized Characteristic Polynomials of Join Graphs and Their ApplicationsPengli Lu0Ke Gao1Yang Yang2School of Computer and Communication, Lanzhou University of Technology, Lanzhou, Gansu 730050, ChinaSchool of Computer and Communication, Lanzhou University of Technology, Lanzhou, Gansu 730050, ChinaSchool of Computer and Communication, Lanzhou University of Technology, Lanzhou, Gansu 730050, ChinaThe Kirchhoff index of G is the sum of resistance distances between all pairs of vertices of G in electrical networks. LEL(G) is the Laplacian-Energy-Like Invariant of G in chemistry. In this paper, we define two classes of join graphs: the subdivision-vertex-vertex join G1⊚G2 and the subdivision-edge-edge join G1⊝G2. We determine the generalized characteristic polynomial of them. We deduce the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomials of G1⊚G2 and G1⊝G2 when G1 is r1-regular graph and G2 is r2-regular graph. As applications, the Laplacian spectra enable us to get the formulas of the number of spanning trees, Kirchhoff index, and LEL of G1⊚G2 and G1⊝G2 in terms of the Laplacian spectra of G1 and G2.http://dx.doi.org/10.1155/2017/2372931
collection DOAJ
language English
format Article
sources DOAJ
author Pengli Lu
Ke Gao
Yang Yang
spellingShingle Pengli Lu
Ke Gao
Yang Yang
Generalized Characteristic Polynomials of Join Graphs and Their Applications
Discrete Dynamics in Nature and Society
author_facet Pengli Lu
Ke Gao
Yang Yang
author_sort Pengli Lu
title Generalized Characteristic Polynomials of Join Graphs and Their Applications
title_short Generalized Characteristic Polynomials of Join Graphs and Their Applications
title_full Generalized Characteristic Polynomials of Join Graphs and Their Applications
title_fullStr Generalized Characteristic Polynomials of Join Graphs and Their Applications
title_full_unstemmed Generalized Characteristic Polynomials of Join Graphs and Their Applications
title_sort generalized characteristic polynomials of join graphs and their applications
publisher Hindawi Limited
series Discrete Dynamics in Nature and Society
issn 1026-0226
1607-887X
publishDate 2017-01-01
description The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices of G in electrical networks. LEL(G) is the Laplacian-Energy-Like Invariant of G in chemistry. In this paper, we define two classes of join graphs: the subdivision-vertex-vertex join G1⊚G2 and the subdivision-edge-edge join G1⊝G2. We determine the generalized characteristic polynomial of them. We deduce the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomials of G1⊚G2 and G1⊝G2 when G1 is r1-regular graph and G2 is r2-regular graph. As applications, the Laplacian spectra enable us to get the formulas of the number of spanning trees, Kirchhoff index, and LEL of G1⊚G2 and G1⊝G2 in terms of the Laplacian spectra of G1 and G2.
url http://dx.doi.org/10.1155/2017/2372931
work_keys_str_mv AT penglilu generalizedcharacteristicpolynomialsofjoingraphsandtheirapplications
AT kegao generalizedcharacteristicpolynomialsofjoingraphsandtheirapplications
AT yangyang generalizedcharacteristicpolynomialsofjoingraphsandtheirapplications
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