On the Estrada and Laplacian Estrada indices of graphs

Abstract The Estrada index of a graph G is defined as EE ( G ) = ∑ i = 1 n e λ i , where λ 1 , λ 2 , … , λ n are the eigenvalues of G. The Laplacian Estrada index of a graph G is defined as LEE ( G ) = ∑ i = 1 n e μ i , where μ 1 , μ 2 , … , μ n are the Laplacian eigenvalues of G. An edge grafting operation on a graph moves a pendent edge between two pendent paths. We study the change of Estrada index of graph under edge grafting operation between two pendent paths at two adjacent vertices. As the application, we give the result on the change of Laplacian Estrada index of bipartite graph under edge grafting operation between two pendent paths at the same vertex. We also determine the unique tree with minimum Laplacian Estrada index among the set of trees with given maximum degree, and the unique trees with maximum Laplacian Estrada indices among the set of trees with given diameter, number of pendent vertices, matching number, independence number and domination number, respectively.

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