39 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
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
Computing Reformulated First Zagreb Index of Some Chemical Graphs as an Application of Generalized Hierarchical Product of Graphs
The generalized hierarchical product of graphs was introduced by L.
Barri\'ere et al in 2009. In this paper, reformulated first Zagreb index of
generalized hierarchical product of two connected graphs and hence as a special
case cluster product of graphs are obtained. Finally using the derived results
the reformulated first Zagreb index of some chemically important graphs such as
square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer
fullerene, zig-zag polyhex nanotube and dicentric dendrimers are computed.Comment: 12 page
On the third ABC index of trees and unicyclic graphs
Let be a simple connected graph with vertex set and edge set
. The third atom-bond connectivity index, index, of is
defined as , where eccentricity is the
largest distance between and any other vertex of , namely
. This work determines the maximal index
of unicyclic graphs with any given girth and trees with any given diameter, and
characterizes the corresponding graphs
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
Corrigendum on Wiener index, Zagreb Indices and Harary index of Eulerian graphs
In the original article ``Wiener index of Eulerian graphs'' [Discrete Applied
Mathematics Volume 162, 10 January 2014, Pages 247-250], the authors state that
the Wiener index (total distance) of an Eulerian graph is maximized by the
cycle. We explain that the initial proof contains a flaw and note that it is a
corollary of a result by Plesn\'ik, since an Eulerian graph is
-edge-connected. The same incorrect proof is used in two referencing papers,
``Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs'' [Bull.
Malays. Math. Sci. Soc. (2019) 42:67-78] and ``Harary index of Eulerian
graphs'' [J. Math. Chem., 59(5):1378-1394, 2021]. We give proofs of the main
results of those papers and the -edge-connected analogues.Comment: 5 Pages, 1 Figure Corrigendum of 3 papers, whose titles are combine