Spectra of Subdivision Vertex-Edge Join of Three Graphs

Fei Wen, You Zhang and Muchun Li
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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

Mathematics, 2019, vol. 7, issue 2, 1-19

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.

Keywords: subdivision vertex-edge join; cospectral mate; spanning tree; Kirchhoff index; Kemeny’s constant (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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