Some Results Of The Laplacian Characteristic Polynomial Of A Graph

If G is a bipartite with n vertices, then the characteristic polynomial σ(G,x) of G has the following form; If G is a bipartite graph, then the energy of G can be expressed by means of the Coulson integral formula which implies that if the numbers of i-matchings of G1and G2satisfy the property m(G1,i)≥m(G2,i) for i=1,2,…,[n/2], i.e., G1≥G2=â†'E(G1)≥E(G2) and G1>G2=â†'E(G1)> E(G2).Note that the characteristic polynomial of adjacency matrix of subdivision graph of a graph G is related to the characteristic polynomial of signless laplacian matrix of graph G as follows: Based on the above result, in the second chapter of this paper, we characterize the graph with chromatic number k which has the maximal coefficients of signless Laplacian characteristic polynomial. So we show that, among all subdivisions of graphs with n vertices and chromatic number k, the subdivision of the Turan graph has the maximal energy.Let G(a1, a2,…, ak) be a simple graph with vertex set V(G)=V1∪V2∪…∪Vk and edge set E(G)={(u,v)|u∈Vi,v∈Vi+1,i=1,2,…, k-1}, where|Vi|=αi>0for1≤i≤k and Vi∩Vj=O for i≠j. Given two positive integers k and n, and k-2positive rational numbers t2,t3,…,t[k/2] and where N is the set of positive integers. In the third chapter of this paper, we prove that all graphs in are cospectral with respect to the normalized Laplacian if it is not an empty set.