61,914 research outputs found
The Connectivity and the Harary Index of a Graph
The Harary index of a graph is defined as the sum of reciprocals of distances
between all pairs of vertices of the graph. In this paper we provide an upper
bound of the Harary index in terms of the vertex or edge connectivity of a
graph. We characterize the unique graph with maximum Harary index among all
graphs with given number of cut vertices or vertex connectivity or edge
connectivity. In addition we also characterize the extremal graphs with the
second maximum Harary index among the graphs with given vertex connectivity
Eccentric connectivity index
The eccentric connectivity index is a novel distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. It is defined as \,, where and
denote the vertex degree and eccentricity of \,, respectively. We survey
some mathematical properties of this index and furthermore support the use of
eccentric connectivity index as topological structure descriptor. We present
the extremal trees and unicyclic graphs with maximum and minimum eccentric
connectivity index subject to the certain graph constraints. Sharp lower and
asymptotic upper bound for all graphs are given and various connections with
other important graph invariants are established. In addition, we present
explicit formulae for the values of eccentric connectivity index for several
families of composite graphs and designed a linear algorithm for calculating
the eccentric connectivity index of trees. Some open problems and related
indices for further study are also listed.Comment: 25 pages, 5 figure
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
On Maximum Signless Laplacian Estrada Indices of Graphs with Given Parameters
Signless Laplacian Estrada index of a graph , defined as
, where are the
eigenvalues of the matrix . We
determine the unique graphs with maximum signless Laplacian Estrada indices
among the set of graphs with given number of cut edges, pendent vertices,
(vertex) connectivity and edge connectivity.Comment: 14 pages, 3 figure
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