# The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs

## TL;DR

This paper studies the properties and bounds of the least H-eigenvalue of the signless Laplacian tensor in connected non-odd-bipartite hypergraphs, focusing on structural changes and extremal hypergraphs.

## Contribution

It provides new insights into the structural properties of first eigenvectors, eigenvalue behavior under branch relocation, and characterizes hypergraphs with minimal least eigenvalues.

## Key findings

- First eigenvectors contain structural information about hypergraphs.
- Relocating odd-bipartite branches affects the least eigenvalue.
- Zero is the infimum of the least eigenvalues for these hypergraphs.

## Abstract

Let $G$ be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of $G$ is simply called the least eigenvalue of $G$ and the corresponding H-eigenvectors are called the first eigenvectors of $G$. In this paper we give some numerical and structural properties about the first eigenvectors of $G$ which contains an odd-bipartite branch, and investigate how the least eigenvalue of $G$ changes when an odd-bipartite branch attached at one vertex is relocated to another vertex. We characterize the hypergraph(s) whose least eigenvalue attains the minimum among a certain class of hypergraphs which contain a fixed non-odd-bipartite connected hypergraph. Finally we present some upper bounds of the least eigenvalue and prove that zero is the least limit point of the least eigenvalues of connected non-odd-bipartite hypergraphs.