On the Seidel energy of uniform hypergraphs due to hyperedge and 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 $m,n \ge 3$.