Signless Laplacian Eigenvalue Of Graphs

For a given graph G, let V(G), E(G), Δ(G),δ(G) denote the vertex-set, edge-set, maximum degree, minimum degree, respectively. The adjacency matrix of G is defined to be the n×n matrix A=(αij)n×n, where αij=1if vi is adjacency to vj;otherwise αij=0. The signless Laplacian matrix Q=D+A where D=diag(d1, d2,…,dn) is the diagonal matrix of vertex degrees and A is the adjacency matrix. It is well known that Q(G) is symmetric matrix. Denote its eigenvalues by q1≥q2≥…≥qn.This master thesis mainly discuss the signless Laplacian eigenvalue of graphs. In chapter1, we give a brief survey in this direction. Main results obtained in this thesis are stated. In chapter2, we study the forth and fifth largest signless Laplacian eigenvalues, prove that the lower bound of them. In chapter3, we give several results about the largest signless Laplacian eigenvalue.