Some Applications Of Spectral Moment

Let G be a simple graph,the adjacency matrix of G,denoted by A(G).The eigenvalues of the graph G,denoted byil,is the eigenvalues of A(G).Let be the k th spectral moment of the graph G,()(())k k k iM G(28) ?l(28)tr A G.In other words, is equal to the number of closed walks of length k in G,and circles of length k is a special kind of closed walks of length k;the k th spectral moment of a vertex is equal to the number of closed walks of length k at the vertex as the starting vertex and ending vertex.Let be the Estrada index of the graph G,()()!ikM GEE G e kl(28) ?(28) ?.According to the formula,comparing the size of the Estrada index of the graph G can transform comparing the size of the spectral moment of the graph G.In this thesis,we will pay our attention on two types of applications of the spectral moment.The main results of the thesis outlined as follows:1.we research applications of spectral moment in calculating the number of circles,then the calculating formula is obtained that the number of circles of length 4 and 5.2.Through researching the spectral moment of a vertex,a new method is obtained that comparing the size of the spectral moment between two vertices;and a new method is obtained that comparing the size of the spectral moment between two graphs.3.Through researching the spectral moment of the graph,we determine the tree graph with the fourth minimum Estrada index,and we determine the unicyclic graph with the second maximum Estrada index.