Graphs with the maximal Estrada indices

Two new transformations are proposed to compare the Estrada indices between two graphs. Let Ψn,mΨn,m be the set of the (n,m)(n,m)-graphs, where n and m   are the numbers of vertices and edges, respectively. The graphs with the maximal Estrada indices in Ψn,mΨn,m are deduced by the new method for three cases, namely unicyclic and bipartite unicyclic graphs (m=nm=n), bicyclic graphs (m=n+1m=n+1), and the (n,m)(n,m)-graphs without even cycles (n+1⩽m⩽3(n−1)/2n+1⩽m⩽3(n−1)/2).