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Mathematics 2019, 7(2), 171; https://doi.org/10.3390/math7020171

Spectra of Subdivision Vertex-Edge Join of Three Graphs

Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China
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Received: 18 January 2019 / Revised: 2 February 2019 / Accepted: 6 February 2019 / Published: 13 February 2019
(This article belongs to the Section Mathematics and Computers Science)
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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. View Full-Text
Keywords: subdivision vertex-edge join; cospectral mate; spanning tree; Kirchhoff index; Kemeny’s constant subdivision vertex-edge join; cospectral mate; spanning tree; Kirchhoff index; Kemeny’s constant
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Wen, F.; Zhang, Y.; Li, M. Spectra of Subdivision Vertex-Edge Join of Three Graphs. Mathematics 2019, 7, 171.

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