39 research outputs found

    Eccentric connectivity index

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    The eccentric connectivity index ΞΎc\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 ΞΎc(G)=βˆ‘v∈V(G)deg(v)β‹…Ο΅(v)\xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \epsilon (v)\,, where deg(v)deg (v) and Ο΅(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

    The comparison of two Zagreb-Fermat eccentricity indices

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    In this paper, we focus on comparing the first and second Zagreb-Fermat eccentricity indices of graphs. We show that βˆ‘uv∈E(G)Ξ΅3(u)Ξ΅3(v)m(G)β‰€βˆ‘u∈V(G)Ξ΅32(u)n(G)\frac{\sum_{uv\in E\left( G \right)}{\varepsilon_3\left( u \right) \varepsilon_3\left( v \right)}}{m\left( G \right)} \leq \frac{\sum_{u\in V\left( G \right)}{\varepsilon_{3}^{2}\left( u \right)}}{n\left( G \right)} 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

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    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

    Graph entropy and related topics

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    On the third ABC index of trees and unicyclic graphs

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    Let G=(V,E)G=(V,E) be a simple connected graph with vertex set V(G)V(G) and edge set E(G)E(G). The third atom-bond connectivity index, ABC3ABC_3 index, of GG is defined as ABC3(G)=βˆ‘uv∈E(G)e(u)+e(v)βˆ’2e(u)e(v)ABC_3(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{e(u)+e(v)-2}{e(u)e(v)}}, where eccentricity e(u)e(u) is the largest distance between uu and any other vertex of GG, namely e(u)=max⁑{d(u,v)∣v∈V(G)}e(u)=\max\{d(u,v)|v\in V(G)\}. This work determines the maximal ABC3ABC_3 index of unicyclic graphs with any given girth and trees with any given diameter, and characterizes the corresponding graphs

    On eccentric connectivity index

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    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

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    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 22-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 22-edge-connected analogues.Comment: 5 Pages, 1 Figure Corrigendum of 3 papers, whose titles are combine
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