82 research outputs found
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
Graph Classes (Dis)satisfying the Zagreb Indices Inequality
International audience{Recently Hansen and Vukicevic proved that the inequality , where and are the first and second Zagreb indices, holds for chemical graphs, and Vukicevic and Graovac proved that this also holds for trees. In both works is given a distinct counterexample for which this inequality is false in general. Here, we present some classes of graphs with prescribed degrees, that satisfy : Namely every graph whose degrees of vertices are in the interval for some integer satisies this inequality. In addition, we prove that for any , there is an infinite family of graphs of maximum degree such that the inequality is false. Moreover, an alternative and slightly shorter proof for trees is presented, as well\ as for unicyclic graphs
The comparison of two Zagreb-Fermat eccentricity indices
In this paper, we focus on comparing the first and second Zagreb-Fermat
eccentricity indices of graphs. We show that holds for all acyclic and unicyclic graphs.
Besides, we verify that the inequality may not be applied to graphs with at
least two cycles
- …