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On the Seidel energy of uniform hypergraphs due to hyperedge and vertex deletion

Shib Sankar Saha

Abstract

Let $\mathcal{S}(\mathcal{H})$ be the Seidel matrix of a hypergraph $\mathcal{H}$, and the Seidel energy is denoted by the sum of the absolute eigenvalues of $\mathcal{S}(\mathcal{H})$. In [G.~X.~Tian, Y.~Li and S.~Y.~Cui, The change of Seidel energy of tripartite Turán graph due to edge deletion, Linear Multilinear Algebra, 19 (2022), 4597-4614] and [Y.~Liu, X.~Chen, The change of Seidel energy of 5-partite Turán graph due to edge deletion, Discrete Applied Mathematics, 2024, 342, 104-123], the authors studied the change of Seidel energy of the Turán graph due to edge deletion. In this article, we analyze the Seidel spectrum of the complete $3$-uniform bipartite hypergraph $\mathcal{C}^3_{m,n}$ and show that it has exactly one negative Seidel eigenvalue even after a single hyperedge deletion. Finally, we prove that the Seidel energy of the complete $3$-uniform bipartite hypergraph $\mathcal{C}^3_{m,n}$ decreases after single hyperedge and vertex deletion for all $m,n \ge 3$.

On the Seidel energy of uniform hypergraphs due to hyperedge and vertex deletion

Abstract

Let be the Seidel matrix of a hypergraph , and the Seidel energy is denoted by the sum of the absolute eigenvalues of . In [G.~X.~Tian, Y.~Li and S.~Y.~Cui, The change of Seidel energy of tripartite Turán graph due to edge deletion, Linear Multilinear Algebra, 19 (2022), 4597-4614] and [Y.~Liu, X.~Chen, The change of Seidel energy of 5-partite Turán graph due to edge deletion, Discrete Applied Mathematics, 2024, 342, 104-123], the authors studied the change of Seidel energy of the Turán graph due to edge deletion. In this article, we analyze the Seidel spectrum of the complete -uniform bipartite hypergraph and show that it has exactly one negative Seidel eigenvalue even after a single hyperedge deletion. Finally, we prove that the Seidel energy of the complete -uniform bipartite hypergraph decreases after single hyperedge and vertex deletion for all .
Paper Structure (4 sections, 11 theorems, 28 equations, 1 figure)

This paper contains 4 sections, 11 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Monotonicity of the Seidel energy under hyperedge deletion
  3. Seidel spectrum of $\mathcal{C}^3_{m,n}$ and $\mathcal{C}^3_{m,n}-e$.
  4. Monotonicity of the Seidel energy under vertex deletion

Key Result

Lemma 2.4

B2012 Let $Q^M$ be the quotient matrix of any square matrix $M$ corresponding to an equitable partition. Then the spectrum of $M$ contains the spectrum of $Q^M$.

Figures (1)

  • Figure 1: Comparison of Seidel energy of a hypergraph due to a strong vertex deletion

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • Lemma 2.10
  • ...and 10 more