186 research outputs found

    The quotients between the (revised) Szeged index and Wiener index of graphs

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    Let Sz(G),Sz∗(G)Sz(G),Sz^*(G) and W(G)W(G) be the Szeged index, revised Szeged index and Wiener index of a graph G.G. In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order n⩾10n\geqslant 10 are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on Sz(G)/W(G)Sz(G)/W(G) is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on Sz∗(G)/W(G)Sz^*(G)/W(G) is identified for GG containing at least one cycle.Comment: 25 pages, 5 figure

    Eccentric connectivity index

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    The eccentric connectivity index ξc\xi^c 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 ξc(G)=∑v∈V(G)deg(v)⋅ϵ(v)\xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \epsilon (v)\,, where deg(v)deg (v) and ϵ(v)\epsilon (v) denote the vertex degree and eccentricity of vv\,, 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
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