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Some Observations on Comparing Zagreb Indices

TL;DR: In this article, the authors search for graphs for which M1/n = M2/m, and show how numerous such graphs can be constructed, which are counterexamples for the earlier conjectured inequality M 1/n ≤ M 2/m.
Abstract: Let G be a simple graph possessing n vertices and m edges. Let di be the degree of the i-th vertex of G, i =1 ,...,n . The first Zagreb index M1 is the sum of d 2 i over all vertices of G . The second Zagreb index M2 is the sum di dj over pairs of adjacent vertices of G . In this paper we search for graph for which M1/n = M2/m , and show how numerous such graphs can be constructed. In addition, we find examples of graphs for which M1/n > M2/m , which are counterexamples for the earlier conjectured inequality M1/n ≤ M2/m .
Citations
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01 Jan 2012
TL;DR: The first Zagreb index of a graph G, with vertex set V (G) and edge set E(G), is defined as M1(G ) = � u∈V(G) d(u) 2 where d denotes the degree of the vertex v as mentioned in this paper.
Abstract: The first Zagreb index of a graph G, with vertex set V (G) and edge set E(G), is defined as M1(G )= � u∈V (G) d(u) 2 where d(u) denotes the degree of the vertex v .A n alternative �

139 citations


Cites background from "Some Observations on Comparing Zagr..."

  • ...A detailed bibliography on recent research of Zagreb indices is found in [4, 19]....

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Journal ArticleDOI
TL;DR: It is shown that this difference of M 1 and M 2 is closely related to the vertex-degree-based invariant R M 2 = ∑ u v ( d u − 1 ) ( d v − 1) , and a few basic properties of R M2 are determined.

96 citations

Journal ArticleDOI
TL;DR: The first and second reformulated Zagreb indices are defined respectively in terms of edge-degrees as EM"1(G)=@?"e"@?"Edeg(e)^2 and EM"2(G)=@?" e"~"fdeg( e)deg(f), where deg(E) denotes the degree of the edge e, and e~f means that the edges e and f are adjacent.

43 citations

01 Jan 2014
TL;DR: In this article, the authors characterized the extremal properties of the first reformulated Zagreb index and determined extremal acyclic and bicyclic graphs with minimum and maximum values of the index by a unified method.
Abstract: The authors Milicevic et al. introduced the reformulated Zagreb indices (19), which is a generalization of classical Zagreb indices of chemical graph theory. In the paper, we characterize the extremal properties of the first reformulated Zagreb index. We first introduce some graph operations which increase or decrease this index. Furthermore, we will determine the extremal acyclic and bicyclic graphs with minimum and maximum of the first Zagreb index by a unified method, respectively. Recently, Ilic and Zhou (18) characterized the extremal graph of unicyclic graphs with the first reformulated Zagreb index. We will provide a shorter proof.

31 citations

Journal Article
TL;DR: In this article, the first and second Zagreb indices of a simple graph G = (V, E) are defined as M1(G ) = � u∈V d 2 =� uv∈E [du + dv ]a nd M2(G ), where du denotes the degree of vertex u.
Abstract: Let G = (V, E) be a simple graph with n = |V | vertices and m = |E| edges. The first and the second Zagreb indices of G are defined as M1(G )= � u∈V d 2 = � uv∈E [du + dv ]a nd M2(G )= � uv∈E du dv, respectively, where du denotes the degree of vertex u .W e co mpare the multiplicative versions of these indices.

25 citations


Cites background from "Some Observations on Comparing Zagr..."

  • ...As in [1] is maintained, the inequality (4) holds for trees [15], unicycles graphs [14], and graphs of maximum degree four, so called molecular graphs [11], graphs with only two distinct vertex degrees, but does not hold in general ( [1, 11, 13])....

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References
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Book
01 Jan 2002
TL;DR: This Users guide notations acronyms list of molecular descriptors contains abbreviations for molecular descriptor values that are useful for counting and topological descriptors calculation.
Abstract: Users guide notations acronyms list of molecular descriptors. Appendices: counting and topological descriptors calculation of descriptors tables of molecular descriptor values.

3,220 citations

Journal ArticleDOI

3,019 citations


"Some Observations on Comparing Zagr..." refers background in this paper

  • ...The second Zagreb index M2 is the sum di dj over pairs of adjacent vertices of G ....

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Journal ArticleDOI
TL;DR: In this paper, the structural dependence of the Huckel total φ-electron energy on the molecular topology of conjugated molecules has been studied and general rules governing the structural properties of the φ energy in conjugate molecules have been derived.

1,706 citations

Book
01 Jan 2009
TL;DR: This book presents a meta-modelling framework for QSAR/QSPR Modeling using Greek alphabets, selected from 450 journals and covering the period from the beginning of molecular descriptor research until the year 2008.
Abstract: Volume I: ALPHABETICAL LISTING Introduction Historical Perspective QSAR/QSPR Modeling How to Learn From This Book Users Guide Notations and Symbols Alphabetical Listing of approx. 3300 entries Greek Alphabet Entries Numerical Entries Volume II: APPENDICES, REFERENCES Full bibliography of more than 6000 references, selected from 450 journals and covering the period from the beginning of molecular descriptor research until the year 2008 Greek alphabets Acronyms Molecular structures

927 citations

Book ChapterDOI
TL;DR: In this paper, the wave function for a π-electron is presented in the LCAO form, where π denotes a p π -vorbital located on the j-th atom of a conjugated molecule, and the summation goes over all n atoms which participate in the conjugation.
Abstract: In the present chapter, as well as throughout the entire book, we assume that the reader knows the basic facts about the Huckel molecular orbital (HMO) theory [35, 51, 62]. Hence HMO theory is an approximate quantum-mechanical approach to the description of the π-electrons in unsaturated conjugated molecules. The wave function for a π-electron is presented in the LCAO form $$ {\psi_i} = \sum\limits_{{j = 1}}^n {{c_{{ij}}}} \left| {{p_j} >} \right. $$ (1) where {p j > symbolizes a p π -vorbital located on the j-th atom of the conjugated molecule, and the summation goes over all n atoms which participate in the conjugation.

797 citations