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

TL;DR

This paper investigates how the Seidel energy of a complete 3-uniform bipartite hypergraph changes when hyperedges or vertices are deleted, revealing that the energy decreases after such deletions.

Contribution

It analyzes the Seidel spectrum of complete 3-uniform bipartite hypergraphs and demonstrates the energy decreases with hyperedge or vertex deletion.

Findings

Seidel spectrum has exactly one negative eigenvalue after deletion.

Seidel energy decreases after hyperedge deletion.

Seidel energy decreases after vertex deletion.

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 Tur\'an 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 Tur\'an graph due to edge deletion, Discrete Applied Mathematics, 2024, 342, 104-123], the authors studied the change of Seidel energy of the Tur\'an 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 $m,n \ge 3$.