On eccentric connectivity index

The eccentric connectivity index, proposed by Sharma, Goswami and Madan, has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature. We now report mathematical properties of the eccentric connectivity index. We establish various lower and upper bounds for the eccentric connectivity index in terms of other graph invariants including the number of vertices, the number of edges, the degree distance and the first Zagreb index. We determine the n-vertex trees of diameter with the minimum eccentric connectivity index, and the n-vertex trees of pendent vertices, with the maximum eccentric connectivity index. We also determine the n-vertex trees with respectively the minimum, second-minimum and third-minimum, and the maximum, second-maximum and third-maximum eccentric connectivity indices forComment: 18 pages, 2 figure

Upper bounds for the sum of Laplacian eigenvalues of graphs

AbstractLet G be a graph with n vertices and e(G) edges, and let μ1(G)⩾μ2(G)⩾⋯⩾μn(G)=0 be the Laplacian eigenvalues of G. Let Sk(G)=∑i=1kμi(G), where 1⩽k⩽n. Brouwer conjectured that Sk(G)⩽e(G)+k+12 for 1⩽k⩽n. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for Sk(G), and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs

On maximum Estrada indices of graphs with given parameters

AbstractFor a graph G with eigenvalues λ1,λ2,…,λn, its Estrada index is defined as EE(G)=∑i=1neλi. We determine the unique graphs with maximum Estrada indices among graphs with given number of cut vertices, connectivity, and edge connectivity, respectively