ON THE SPECTRA OF TENSOR JOIN OF HYPERGRAPHS

Document Type : Original Manuscript

Authors

Department of Mathematics, The Gandhigram Rural Institute (Deemed to be University), Gandhigram–624 302, Tamil Nadu, India.

10.22044/jas.2025.14919.1880

Abstract

In this paper, we consider certain classes of hypergraphs constructed from the tensor join of hypergraphs, specifically the tensor join of hypergraphs constrained by vertex subsets and the $(H, \mathcal{T}_{\mathcal{S}})$-join of hypergraphs constrained by $\mathcal{S}$. We determine some eigenvalues of the adjacency tensor of these classes of hypergraphs by establishing corresponding eigenvectors. We demonstrate that the eigenvalues of the adjacency tensor of the constituting hypergraphs are also eigenvalues of the adjacency tensor of the join of a set of hypergraphs. Furthermore, as a special case of our results, we provide some eigenvalues and eigenvectors of the adjacency tensor for the join of non-uniform hypergraphs on a backbone hypergraph $H$ (and, similarly, for the join of $m$-uniform hypergraphs on a backbone hypergraph $H$). Additionally, we establish a relationship between the eigenvalues of the adjacency tensor of a hypergraph $H$ and certain eigenvalues of the adjacency tensor of the $(H, \mathcal{T}_{\mathcal{S}})$-join of hypergraphs constrained by $\mathcal{S}$. Using this relationship, we determine some eigenvalues and their corresponding eigenvectors for the adjacency tensor of the lexicographic product of two hypergraphs.

Keywords

References

1. A. Banerjee, On the spectrum of hypergraphs, Linear Algebra Appl., 614 (2021), 82–110.
2. A. Banerjee, A. Char and B. Mondal, Spectra of general hypergraphs, Linear Algebra Appl., 518 (2017), 14–30.
3. J. Cooper and A. Dutle, Spectra of uniform hypergraphs, Linear Algebra Appl., 436(9) (2012), 3268–3292.
4. M. Hellmuth, L. Ostermeier and P. F. Stadler, A survey on hypergraph products, Math. Comput. Sci., 6 (2012), 1–32.
5. S. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph, J. Comb. Optim., 24(4) (2012), 564–579.
6. L. Kang, L. Liu and E. Shan, The eigenvectors to the p-spectral radius of general hypergraphs, J. Comb. Optim., 38 (2019), 556–569.
7. G. Li, L. Qi and G. Yu, The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory, Numer. Linear Algebra Appl., 20(6) (2013), 1001–1029.
8. H. Lia and C. Dengb, On the principal eigenvectors of general hypergraphs, ScienceAsia, 50(2) (2024), Article ID: 2024006.
9. K. Pearson and T. Zhang, Eigenvalues of the adjacency tensor on products of hypergraphs, Int. J. Contemp. Math. Sciences, 8(4) (2013), 151–158.
10. L. Qi, H+-eigenvalues of Laplacian tensor and signless Laplacians, Commun. Math. Sci., 12 (2014), 1045–1064.
11. L. Qi and Z. Luo, Tensor Analysis: Spectral theory and special tensors, SIAM, Philadelphia, 2017.
12. S. Rota Bulò and M. Pelillo, New bounds on the clique number of graphs based on spectral hypergraph theory, International Conference on Learning and Intelligent Optimization, Berlin, Heidelberg: Springer Berlin Heidelberg, (2009), 45–58.
13. A. Sarkar and A. Banerjee, Joins of hypergraphs and their spectra, Linear Algebra Appl., 603 (2020), 101–129.
14. R. Vishnupriya and R. Rajkumar, Tensor join of hypergraphs and its spectra, Le Matematiche, 78(1) (2023), 111–147.
15. J. Xie and A. Chang, On the Z-eigenvlues of the signless Laplacian tensor or an even uniform hypergraph, Numer. Linear Algebra Appl., 20(6) (2013), 1030–1045.
16. W. Zhang, L. Liu, L. Kang and Y. Bai, Some properties of the spectral radius for general hypergraphs, Linear Algebra Appl., 513 (2017), 103–119. 
Journal of Algebraic Systems
Volume 14, Issue 3
June 2026
Pages 489-508

Files

Share

How to cite

Statistics