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Spectra of Subdivision-Vertex Join and Subdivision-Edge Join of Two Graphs

  • Xiaogang Liu
  • Zuhe Zhang
Article

Abstract

The subdivision graph \({\mathcal {S}}(G)\) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let \(G_1\) and \(G_2\) be two vertex disjoint graphs. The subdivision-vertex join of \(G_1\) and \(G_2\), denoted by \(G_1{\dot{\vee }}G_2\), is the graph obtained from \({\mathcal {S}}(G_1)\) and \(G_2\) by joining every vertex of \(V(G_1)\) with every vertex of \(V(G_2)\). The subdivision-edge join of \(G_1\) and \(G_2\), denoted by \(G_1{\underline{\vee }}G_2\), is the graph obtained from \({\mathcal {S}}(G_1)\) and \(G_2\) by joining every vertex of \(I(G_1)\) with every vertex of \(V(G_2)\), where \(I(G_1)\) is the set of inserted vertices of \({\mathcal {S}}(G_1)\). In this paper, we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of \(G_1{\dot{\vee }}G_2\) (respectively, \(G_1{\underline{\vee }}G_2\)) for a regular graph \(G_1\) and an arbitrary graph \(G_2\), in terms of the corresponding spectra of \(G_1\) and \(G_2\). As applications, these results enable us to construct infinitely many pairs of cospectral graphs. We also give the number of the spanning trees and the Kirchhoff index of \(G_1{\dot{\vee }}G_2\) (respectively, \(G_1{\underline{\vee }}G_2\)) for a regular graph \(G_1\) and an arbitrary graph \(G_2\).

Keywords

Spectrum Cospectral graphs Subdivision-vertex join Subdivision-edge join Spanning tree Kirchhoff index 

Mathematics Subject Classification

05C50 

Notes

Acknowledgements

The authors appreciate the anonymous referees for their comments and suggestions.

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia

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