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The spectrum of the Corona of Hypergraphs

Liya Jess Kurian, Chithra A.

Abstract

The corona of hypergraphs is an extension of the corona operation applied to graphs. The corona $G_0^* \odot_1^n G_1^*$ of two hypergraphs is obtained by taking $n$ copies of $G_1^*$ (where $n$ is the order of $G_0^*$) and by joining the $i$-th vertex of $G_0^*$ with the $i$-th copy of $G_1^*$. In this paper, we estimate the complete spectrum(adjacency and Seidel) and eigenvectors of the corona $G_0^* \odot_1^n G_1^*$ of two hypergraphs when $G_1^*$ is regular. Additionally, we define the corona hypergraph $G_0^{*(m)}=G_0^{*(m-1)} \odot_1^n G_0^*$ and determined its adjacency spectrum. Also, we extend the definition coronal of the adjacency matrix. Moreover, we estimate the characteristic polynomial of Seidel matrix of the generalised corona of hypergraphs. Applying these results, we obtain infinitely many non-regular non-isomorphic adjacency and Seidel cospectral hypergraphs.

The spectrum of the Corona of Hypergraphs

Abstract

The corona of hypergraphs is an extension of the corona operation applied to graphs. The corona of two hypergraphs is obtained by taking copies of (where is the order of ) and by joining the -th vertex of with the -th copy of . In this paper, we estimate the complete spectrum(adjacency and Seidel) and eigenvectors of the corona of two hypergraphs when is regular. Additionally, we define the corona hypergraph and determined its adjacency spectrum. Also, we extend the definition coronal of the adjacency matrix. Moreover, we estimate the characteristic polynomial of Seidel matrix of the generalised corona of hypergraphs. Applying these results, we obtain infinitely many non-regular non-isomorphic adjacency and Seidel cospectral hypergraphs.
Paper Structure (11 sections, 17 theorems, 80 equations, 4 figures)

This paper contains 11 sections, 17 theorems, 80 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalised corona of hypergraphs
  4. Adjacency spectrum of corona of hypergraphs
  5. Seidel Spectrum of the corona of hypergraphs
  6. Spectrum of the corona of two hypergraphs
  7. Coronal of the hypergraph
  8. Spectrum of the corona hypergraph $G_0^{*(m)}$
  9. Construction of Seidel-cospectral hypergraphs
  10. Conclusion
  11. Declarations

Key Result

Lemma 2.1

Das2018 Let $M_{11}, M_{12}, M_{21}$, and $M_{22}$ be matrices with $M_{22}$ invertible. Let Then, $det(M) = det(M_{22})det(M_{11}-M_{12}M_{22}^{-1}M_{21}).$

Figures (4)

  • Figure 1: $K_3^3$
  • Figure 2: $G_0^*$
  • Figure 3: $G^*=G_0^* \odot_1^n K_3^3$
  • Figure 5: $K_3^{3(3)}$

Theorems & Definitions (34)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Theorem 3.1.1
  • proof
  • Corollary 3.1.2
  • proof
  • Theorem 3.2.1
  • ...and 24 more