| Algebraic graph theory is a research direction combining algebra and graph theory.It mainly uses algebraic methods and conclusions to study the main problems in graph theory.Graph spectrum theory is an important branch of algebraic graph theory,which studies graph structure from the perspective of spectrum.According to different matrix representations,graph spectrum theory can be generally divided into adjacency spectrum theory,Laplacian spectrum theory and unsigned Laplacian spectrum theory.In 2000,E.Estrada defined the Estrada index for the study of protein molecular mechanisms.Estrada index is an important concept based on adjacency spectrum of graphs.Similar to the Estrada index,the Laplacian Estrada index is a concept based on Laplacian spectrum of graphs.For Estrada index,people mainly study Estrada index from two perspectives.The first is to estimate the upper and lower bounds of Estrada index,and the second is to study the graphs with extreme Estrada index.Similarly,the Laplacian Estrada index of graphs is also studied.This article studies the bounds of the Estrada index of connected graphs.Regarding the Laplacian Estrada index,study connected graphs with the maximum Laplacian Estrada index.The main work of this paper is as follows:1.We study the bound of the Estrada index on connected graphs.The lower bound of the Estrada index of connected graphs and bipartite graphs is obtained by using the fourth order spectral moment of the path.In addition,by using the special properties of semi-regular graphs,two bounds of the Estrada index of semi-regular graphs are obtained.2.We study the graphs with maximum Laplacian Estrada index in connected graphs.When the number of edges and vertices of graphs are fixed,the upper bound of Laplacian Estrada index is determined,and the structural characteristics of the graphs are characterized when the upper bound is reached. |
