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: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1026-0226 1607-887X |