# The least signless Laplacian eigenvalue of the complements of bicyclic graphs

## TL;DR

This paper characterizes the unique connected graph with the minimum least signless Laplacian eigenvalue among complements of bicyclic graphs using graft transformations.

## Contribution

It introduces two graft transformations and applies them to identify the graph with the minimal eigenvalue in this class.

## Key findings

- Identifies the unique connected graph with minimum least signless Laplacian eigenvalue among complements of bicyclic graphs.
- Develops graft transformations as tools for spectral graph analysis.
- Provides a characterization that can guide future spectral graph theory research.

## Abstract

Suppose that $G$ is a connected simple graph with the vertex set $V(G)=\{v_1, v_2,\cdots,v_n\}$. Then the adjacency matrix of $G$ is $A(G)=(a_{ij})_{n\times n}$, where $a_{ij}=1$ if $v_i$ is adjacent to $v_j$, and otherwise $a_{ij}=0$. The degree matrix $D(G)=diag(d_{G}(v_1), d_{G}(v_2), \dots, d_{G}(v_n)),$ where $d_{G}(v_i)$ denotes the degree of $v_i$ in the graph $G$ ($1\leq i\leq n$). The matrix $Q(G)=D(G)+A(G)$ is called the signless Laplacian matrix of $G$. The least eigenvalue of $Q(G)$ is also called the least signless Laplacian eigenvalue of $G$. In this paper we give two graft transformations and then use them to characterize the unique connected graph whose least signless Laplacian eigenvalue is minimum among the complements of all bicyclic graphs.