scholarly journals On the eccentric connectivity coindex in graphs

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><p>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</p> <p><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></p> <p>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.</p></abstract>

The eccentric adjacency index of unicyclic graphs and trees

2018 ◽  
Vol 13 (01) ◽  
pp. 2050028 ◽  
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.

Extermal forgotten topological index of quasi-unicyclic graphs

2021 ◽  
pp. 2250041
Author(s):  
Amir Taghi Karimi
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Index ◽  
Unicyclic Graph ◽  
Upper And Lower Bounds ◽  
Unicyclic Graphs ◽  
Degree Of A Vertex

The forgotten topological index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] overall edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. The graph [Formula: see text] is called a quasi-unicyclic graph if there exists a vertex [Formula: see text] such that [Formula: see text] is a connected graph with a unique cycle. In this paper, we give sharp upper and lower bounds for the F-index (forgotten topological index) of the quasi-unicyclic graphs.

scholarly journals Physical-chemical properties studying of molecular structures via topological index calculating

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 261-269
Author(s):  
Jianzhang Wu ◽  
Mohammad Reza Farahani ◽  
Xiao Yu ◽  
Wei Gao
Keyword(s):  
Topological Index ◽  
Chemical Properties ◽  
Molecular Structures ◽  
Connectivity Index ◽  
Chemical Characteristics ◽  
Eccentric Connectivity Index ◽  
Distance Computation ◽  
Physical Chemical ◽  
Mathematical Derivation ◽  
Mathematical Standpoint

AbstractIt’s revealed from the earlier researches that many physical-chemical properties depend heavily on the structure of corresponding moleculars. This fact provides us an approach to measure the physical-chemical characteristics of substances and materials. In our article, we report the eccentricity related indices of certain important molecular structures from mathematical standpoint. The eccentricity version indices of nanostar dendrimers are determined and the reverse eccentric connectivity index for V-phenylenic nanotorus is discussed. The conclusions we obtained mainly use the trick of distance computation and mathematical derivation, and the results can be applied in physics engineering.

General eccentric connectivity index of trees and unicyclic graphs

2020 ◽  
Vol 284 ◽  
pp. 301-315
Author(s):  
Tomáš Vetrík ◽  
Mesfin Masre
Keyword(s):  
Connectivity Index ◽  
Eccentric Connectivity Index ◽  
Unicyclic Graphs

General sum-connectivity index of unicyclic graphs with given diameter and girth

2021 ◽  
pp. 2150140
Author(s):  
Tomáš Vetrík
Keyword(s):  
Lower Bounds ◽  
Topological Indices ◽  
Connectivity Index ◽  
Index Formula ◽  
Extremal Graphs ◽  
Unicyclic Graphs ◽  
Harmonic Index

Topological indices of graphs have been studied due to their extensive applications in chemistry. We obtain lower bounds on the general sum-connectivity index [Formula: see text] for unicyclic graphs [Formula: see text] of given girth and diameter, and for unicyclic graphs of given diameter, where [Formula: see text]. We present the extremal graphs for all the bounds. Our results generalize previously known results on the harmonic index for unicyclic graphs of given diameter.

scholarly journals Maximum Reciprocal Degree Resistance Distance Index of Unicyclic Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Gai-Xiang Cai ◽  
Xing-Xing Li ◽  
Gui-Dong Yu
Keyword(s):  
Connected Graph ◽  
Extremal Graph ◽  
Resistance Distance ◽  
Unicyclic Graphs

The reciprocal degree resistance distance index of a connected graph G is defined as RDRG=∑u,v⊆VGdGu+dGv/rGu,v, where rGu,v is the resistance distance between vertices u and v in G. Let Un denote the set of unicyclic graphs with n vertices. We study the graph with maximum reciprocal degree resistance distance index among all graphs in Un and characterize the corresponding extremal graph.

SOME NEW LOWER BOUNDS FOR THE KIRCHHOFF INDEX OF A GRAPH

2017 ◽  
Vol 97 (1) ◽  
pp. 1-10
Author(s):  
I. MILOVANOVIĆ ◽  
M. MATEJIĆ ◽  
E. GLOGIĆ ◽  
E. MILOVANOVIĆ
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Index ◽  
Its Sequence ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let$G$be a simple connected graph with$n$vertices and$m$edges and$d_{1}\geq d_{2}\geq \cdots \geq d_{n}>0$its sequence of vertex degrees. If$\unicode[STIX]{x1D707}_{1}\geq \unicode[STIX]{x1D707}_{2}\geq \cdots \geq \unicode[STIX]{x1D707}_{n-1}>\unicode[STIX]{x1D707}_{n}=0$are the Laplacian eigenvalues of$G$, then the Kirchhoff index of$G$is$\mathit{Kf}(G)=n\sum _{i=1}^{n-1}\unicode[STIX]{x1D707}_{i}^{-1}$. We prove some new lower bounds for$\mathit{Kf}(G)$in terms of some of the parameters$\unicode[STIX]{x1D6E5}=d_{1}$,$\unicode[STIX]{x1D6E5}_{2}=d_{2}$,$\unicode[STIX]{x1D6E5}_{3}=d_{3}$,$\unicode[STIX]{x1D6FF}=d_{n}$,$\unicode[STIX]{x1D6FF}_{2}=d_{n-1}$and the topological index$\mathit{NK}=\prod _{i=1}^{n}d_{i}$.

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

2019 ◽  
Vol 29 (2) ◽  
pp. 193-202 ◽  
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.

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

2018 ◽  
Vol 10 (05) ◽  
pp. 1850065 ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document