Some Results Of The Coefficients Of Laplacian Characteristic Polynomials Of Graphs

Li, Shiu and Chang (The number of spanning trees of a graph, Appl. Math. Lett.,2010,23:286-290) obtained some upper bounds for the number of spanning trees of graphs.In the second chapter of this paper, we study the upper bound of the coefficients ofLaplacian characteristic polynomial of connected graphs with the connectivity, edge-connectivityand chromatic number, and show the structure of graph that reach the upper bound. Since thenumber of spanning trees of a graph is determined by one of the coefficients of the Laplaciancharacteristic polynomial of the graph, we have generalized the results by Li, Shiu, and Chang.Zhou and Gutman (A connection between ordinary and Laplacian spectra of bipartite graphs,Linear Multilinear Algebra,2008,56:305-310) found the connection between the Laplaciancharacteristic polynomial of a bipartite and the characteristic polynomial of its subdivision graph.In the third chapter of this paper, we consider the relationship between the Laplaciancharacteristic polynomial of connected graphs and the characteristic polynomial of itssubdivision graph, we consider whether and when there are differences between the coefficientsof the these two polynomials, and characterize expression of coefficients differences. We findthat the signless Laplacian characteristic polynomials of graph is an important bridge. Thus wealso study its coefficients and prove the combinational explanation of coefficients in a new way.