The least signless Laplacian eigenvalue of the complements of bicyclic graphs

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 $G$ ($1\le i\le n$). 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.