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Spectra of Subdivision Vertex-Edge Join of Three Graphs

Author

Listed:
  • Fei Wen

    (Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China)

  • You Zhang

    (Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China)

  • Muchun Li

    (Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China)

Abstract

In this paper, we introduce a new graph operation called subdivision vertex-edge join (denoted by G 1 S ▹ ( G 2 V ∪ G 3 E ) for short), and then the adjacency spectrum , the Laplacian spectrum and the signless Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) are respectively determined in terms of the corresponding spectra for a regular graph G 1 and two arbitrary graphs G 2 and G 3 . All the above can be viewed as the generalizations of the main results in [X. Liu, Z. Zhang, Bull. Malays. Math. Sci. Soc. , 2017:1–17]. Furthermore, we also determine the normalized Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) whenever G i are regular graphs for each index i = 1 , 2 , 3 . As applications, we construct infinitely many pairs of A-cospectral mates , L-cospectral mates , Q-cospectral mates and L - cospectral mates . Finally, we give the number of spanning trees , the ( degree- ) Kirchhoff index and the Kemeny’s constant of G 1 S ▹ ( G 2 V ∪ G 3 E ) , respectively.

Suggested Citation

  • Fei Wen & You Zhang & Muchun Li, 2019. "Spectra of Subdivision Vertex-Edge Join of Three Graphs," Mathematics, MDPI, vol. 7(2), pages 1-19, February.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:2:p:171-:d:205701
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    References listed on IDEAS

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    1. van Dam, E.R. & Haemers, W.H., 2002. "Which Graphs are Determined by their Spectrum?," Discussion Paper 2002-66, Tilburg University, Center for Economic Research.
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    Cited by:

    1. Jia Wei & Jing Wang, 2022. "Spectra of Complemented Triangulation Graphs," Mathematics, MDPI, vol. 10(17), pages 1-9, September.

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