The Least Signless Laplacian Eigenvalue Of Unicyclic Graphs

Graph spectral theory is a part of the graph theory and combinatorial matrix theory,which has received more and more researchers' attention,and in the quantum chemistry,statistical mechanics,computer science and communication network have been widely used in information science.In recent years,the study of the eigenvalues of the research is very active topic.A large number of articles focus on the figure of signless Laplacian matrix,Laplacian matrix and the study of the nature of adjacency matrix,such as for signless Laplacian eigenvalue research,maximum and minimum eigenvalue of the boundary of research as well as under the disturbation of signless Laplacian,some properties of the smallest eigenvalue.By E.R.Van Dam,W.H Haemers wrote in 2003 "Which graphs are determined by their spectrum "points out that in the distinction between the composition of the signless Laplacian spectrum of a graph than the Laplacian spectrum or adjacency matrix spectrum can better the nature of the response figure,at the same time,the least signless Laplacian eigenvalue of nonbipartite unicyclic graph is an important indicator to judge.Paper[9]gives the graph which the least eigenvalue of signless Laplacian matrix to achieve the minimum,namely the triangle joints a way at one endpoint of the triangle.This paper mainly discusses the least eigenvalue of the signless Laplacian.Given the graph which mixmizing the least eigenvalue of signless Laplacian of all the unicyclic.mainly using the following methods,such as grafting,quadratic,characteristic equation to study the signless Laplacian spectrum deeply.