Several Studies Of Mapping Problem

A spectral graph theory is a theory in which graphs are studied by means of eigen-values of a matrix M which is in a prescribed way defined for any graph. This theoryis called M-theory. Graphs matrices include the adjacency matrix A, the Laplacianmatrix L and signless Laplacian matrix Q and so on. In this thesis, we mainly studythe A-spectral and Q-spectral of a graph.Denote φ(G, λ)=φ(G) be the characteristic polynomial of the adjacency matrixA of G. If φ(G, λ)=φ(H, λ), then G and H are cospectral for A, denoted by G～H.We denoted the cospectral equivalence class by [G]φ=um (or simply G is a {HDS-|φ(H)=φ(G) to determined by its spectrgraph), if H～=}. A graph Gis said G wheneverH～G.Recall that the signless Laplacian matrix Q(G)=D(G)+A(G), where D(G)=diag(d1, d2,, dn) is the corresponding diagonal matrix of vertex degrees.In the first chapter of this paper we introduce some basic concepts as well as theresearch progress of graph theory; in chapter2, we study the spectral characteristicabout A of graphs with the radius less than2and path as its component; in chapter3, We first determine the relation between the Q-spectrum and the circumference ofG. Using this relation we characterize all connected graphs with exactly three Q-eigenvalues at least2and obtain all minimal forbidden subgraphs w.r.t. this property.