New and Modified Eccentric Indices of Octagonal Grid Omn

AbstractThe eccentricity εu of vertex u in a connected graph G, is the distance between u and a vertex farthermost from u. The aim of the present paper is to introduce new eccentricity based index and eccentricity based polynomial, namely modified augmented eccentric connectivity index and modified augmented eccentric connectivity polynomial respectively. As an application we compute these new indices for octagonal grid $\begin{array}{} \displaystyle O_n^m \end{array}$ and we compare the results obtained with the ones obtained by other indices like Ediz eccentric connectivity index, modified eccentric connectivity index and modified eccentric connectivity polynomial ECP(G, x).

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Minimum eccentric connectivity index for graphs with fixed order and fixed number of pendant vertices

Yugoslav journal of operations research ◽

10.2298/yjor181115010d ◽

2019 ◽

Vol 29 (2) ◽

pp. 193-202 ◽

Cited By ~ 1

Author(s):

Gauvain Devillez ◽

Alain Hertz ◽

Hadrien Mélot ◽

Pierre Hauweele

Keyword(s):

Connected Graph ◽

Fixed Number ◽

Connectivity Index ◽

Maximum Distance ◽

Eccentric Connectivity Index ◽

Connected Graphs ◽

Fixed Order ◽

Elegant Proof

The eccentric connectivity index of a connected graph G is the sum over all vertices v of the product dG(v)eG(v), where dG(v) is the degree of v in G and eG(v) is the maximum distance between v and any other vertex of G. We characterize, with a new elegant proof, those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order n. Then, given two integers n and p with p ? n - 1, we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order n with p pendant vertices.

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The eccentric version of atom-bond connectivity index of tetra sheet networks

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500659 ◽

2018 ◽

Vol 10 (05) ◽

pp. 1850065 ◽

Cited By ~ 1

Author(s):

Muhammad Imran ◽

Abdul Qudair Baig ◽

Muhammad Razwan Azhar

Keyword(s):

Connected Graph ◽

Molecular Graph ◽

Theoretical Chemistry ◽

Connectivity Index ◽

Maximum Distance ◽

Eccentric Connectivity Index ◽

Topological Descriptor ◽

Degree Of A Vertex ◽

Connectivity Indices ◽

Atom Bond

Among topological descriptor of graphs, the connectivity indices are very important and they have a prominent role in theoretical chemistry. The atom-bond connectivity index of a connected graph [Formula: see text] is represented as [Formula: see text], where [Formula: see text] represents the degree of a vertex [Formula: see text] of [Formula: see text] and the eccentric connectivity index of the molecular graph [Formula: see text] is represented as [Formula: see text], where [Formula: see text] is the maximum distance between the vertex [Formula: see text] and any other vertex [Formula: see text] of the graph [Formula: see text]. The new eccentric atom-bond connectivity index of any connected graph [Formula: see text] is defined as [Formula: see text]. In this paper, we compute the new eccentric atom-bond connectivity index for infinite families of tetra sheets equilateral triangular and rectangular networks.

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The eccentric adjacency index of unicyclic graphs and trees

Asian-European Journal of Mathematics ◽

10.1142/s179355712050028x ◽

2018 ◽

Vol 13 (01) ◽

pp. 2050028 ◽

Cited By ~ 3

Author(s):

Shehnaz Akhter ◽

Rashid Farooq

Keyword(s):

Connected Graph ◽

Connectivity Index ◽

Eccentric Connectivity Index ◽

Unicyclic Graphs ◽

Vertex Set ◽

Simple Connected Graph ◽

Eccentric Adjacency Index

Let [Formula: see text] be a simple connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. The eccentricity [Formula: see text] of a vertex [Formula: see text] in [Formula: see text] is the largest distance between [Formula: see text] and any other vertex of [Formula: see text]. The eccentric adjacency index (also known as Ediz eccentric connectivity index) of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the sum of degrees of neighbors of the vertex [Formula: see text]. In this paper, we determine the unicyclic graphs with largest eccentric adjacency index among all [Formula: see text]-vertex unicyclic graphs with a given girth. In addition, we find the tree with largest eccentric adjacency index among all the [Formula: see text]-vertex trees with a fixed diameter.

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On the eccentric connectivity coindex in graphs

AIMS Mathematics ◽

10.3934/math.2022041 ◽

2021 ◽

Vol 7 (1) ◽

pp. 651-666

Author(s):

Hongzhuan Wang ◽

◽

Xianhao Shi ◽

Ber-Lin Yu

Keyword(s):

Lower Bounds ◽

Connected Graph ◽

Topological Index ◽

Extremal Problems ◽

Connectivity Index ◽

Adjacent Vertex ◽

Extremal Graph ◽

Eccentric Connectivity Index ◽

Unicyclic Graphs

<abstract>The well-studied eccentric connectivity index directly consider the contribution of all edges in a graph. By considering the total eccentricity sum of all non-adjacent vertex, Hua et al. proposed a new topological index, namely, eccentric connectivity coindex of a connected graph. The eccentric connectivity coindex of a connected graph $ G $ is defined as <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \overline{\xi}^{c}(G) = \sum\limits_{uv\notin E(G)} (\varepsilon_{G}(u)+\varepsilon_{G}(v)). $\end{document} </tex-math></disp-formula> Where $ \varepsilon_{G}(u) $ (resp. $ \varepsilon_{G}(v) $) is the eccentricity of the vertex $ u $ (resp. $ v $). In this paper, some extremal problems on the $ \overline{\xi}^{c} $ of graphs with given parameters are considered. We present the sharp lower bounds on $ \overline{\xi}^{c} $ for general connecteds graphs. We determine the smallest eccentric connectivity coindex of cacti of given order and cycles. Also, we characterize the graph with minimum and maximum eccentric connectivity coindex among all the trees with given order and diameter. Additionally, we determine the smallest eccentric connectivity coindex of unicyclic graphs with given order and diameter and the corresponding extremal graph is characterized as well.</abstract>

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On eccentricity-based topological descriptors of water-soluble dendrimers

Zeitschrift für Naturforschung C ◽

10.1515/znc-2018-0123 ◽

2018 ◽

Vol 74 (1-2) ◽

pp. 25-33 ◽

Cited By ~ 11

Author(s):

Zahid Iqbal ◽

Muhammad Ishaq ◽

Adnan Aslam ◽

Wei Gao

Keyword(s):

Molecular Structure ◽

Chemical Properties ◽

Topological Indices ◽

Water Soluble ◽

Connectivity Index ◽

Topological Descriptors ◽

Physical And Chemical Properties ◽

Eccentric Connectivity Index ◽

Physical And Chemical ◽

And Chemical Properties

AbstractPrevious studies show that certain physical and chemical properties of chemical compounds are closely related with their molecular structure. As a theoretical basis, it provides a new way of thinking by analyzing the molecular structure of the compounds to understand their physical and chemical properties. The molecular topological indices are numerical invariants of a molecular graph and are useful to predict their bioactivity. Among these topological indices, the eccentric-connectivity index has a prominent place, because of its high degree of predictability of pharmaceutical properties. In this article, we compute the closed formulae of eccentric-connectivity–based indices and its corresponding polynomial for water-soluble perylenediimides-cored polyglycerol dendrimers. Furthermore, the edge version of eccentric-connectivity index for a new class of dendrimers is determined. The conclusions we obtained in this article illustrate the promising application prospects in the field of bioinformatics and nanomaterial engineering.

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