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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| Format: | Article |
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Wiley
2017-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2017/2372931 |
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| author | Pengli Lu Ke Gao Yang Yang |
| author_facet | Pengli Lu Ke Gao Yang Yang |
| author_sort | Pengli Lu |
| collection | DOAJ |
| 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. |
| format | Article |
| id | doaj-art-ce381ae1e6e54c0ebedc2f3e795f4ecd |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-ce381ae1e6e54c0ebedc2f3e795f4ecd2025-02-03T06:11:10ZengWileyDiscrete 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 |
| spellingShingle | Pengli Lu Ke Gao Yang Yang Generalized Characteristic Polynomials of Join Graphs and Their Applications Discrete Dynamics in Nature and Society |
| title | 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_short | Generalized Characteristic Polynomials of Join Graphs and Their Applications |
| title_sort | generalized characteristic polynomials of join graphs and their applications |
| url | http://dx.doi.org/10.1155/2017/2372931 |
| work_keys_str_mv | AT penglilu generalizedcharacteristicpolynomialsofjoingraphsandtheirapplications AT kegao generalizedcharacteristicpolynomialsofjoingraphsandtheirapplications AT yangyang generalizedcharacteristicpolynomialsofjoingraphsandtheirapplications |