# 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

05C50

## Notes

### Acknowledgements

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

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