New Results on the Forgotten Topological Index and Coindex

Journal of Mathematics ◽

10.1155/2021/8700736 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Akbar Jahanbani ◽

Maryam Atapour ◽

Rana Khoeilar

Keyword(s):

Electron Energy ◽

Connected Graph ◽

Topological Index ◽

Nonadjacent Vertex ◽

Zagreb Indices ◽

Vertex Degrees ◽

Simple Connected Graph

The ℱ -coindex (forgotten topological coindex) for a simple connected graph G is defined as the sum of the terms ζ G 2 y + ζ G 2 x over all nonadjacent vertex pairs x , y of G , where ζ G y and ζ G x are the degrees of the vertices y and x in G , respectively. The ℱ -index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972 in the same paper where the first and second Zagreb indices were introduced to study the structure dependency of total π -electron energy. Therefore, considering the importance of the ℱ -index and ℱ -coindex, in this paper, we study these indices, and we present new bounds for the ℱ -index and ℱ -coindex.

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Bounds for symmetric division deg index of graphs

Filomat ◽

10.2298/fil1903683d ◽

2019 ◽

Vol 33 (3) ◽

pp. 683-698 ◽

Cited By ~ 1

Author(s):

Kinkar Das ◽

Marjan Matejic ◽

Emina Milovanovic ◽

Igor Milovanovic

Keyword(s):

Connected Graph ◽

Maximum Degree ◽

Topological Index ◽

Minimum Degree ◽

Topological Indices ◽

Symmetric Division ◽

Mathematical Properties ◽

Vertex Degrees ◽

Additive Modeling ◽

Simple Connected Graph

LetG = (V,E) be a simple connected graph of order n (?2) and size m, where V(G) = {1, 2,..., n}. Also let ? = d1 ? d2 ?... ? dn = ? > 0, di = d(i), be a sequence of its vertex degrees with maximum degree ? and minimum degree ?. The symmetric division deg index, SDD, was defined in [D. Vukicevic, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta 83 (2010) 261- 273] as SDD = SDD(G) = ?i~j d2i+d2j/didj, where i~j means that vertices i and j are adjacent. In this paper we give some new bounds for this topological index. Moreover, we present a relation between topological indices of graph.

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SOME NEW LOWER BOUNDS FOR THE KIRCHHOFF INDEX OF A GRAPH

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972717000831 ◽

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$\mathit{NK}=\prod _{i=1}^{n}d_{i}$.

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F-Index of some graph operations

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830916500257 ◽

2016 ◽

Vol 08 (02) ◽

pp. 1650025 ◽

Cited By ~ 18

Author(s):

Nilanjan De ◽

Sk. Md. Abu Nayeem ◽

Anita Pal

Keyword(s):

Electron Energy ◽

Topological Index ◽

Zagreb Index ◽

Zagreb Indices ◽

Molecular Graphs ◽

Basic Properties ◽

Graph Operations ◽

Physico Chemical ◽

Vertex Degrees

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total [Formula: see text]-electron energy. But this topological index was not further studied till then. Very recently, Furtula and Gutman [A forgotten topological index,J. Math. Chem. 53(4) (2015) 1184–1190.] reinvestigated the index and named it “forgotten topological index” or “F-index”. In that paper, they present some basic properties of this index and showed that this index can enhance the physico-chemical applicability of Zagreb index. Here, we study the behavior of this index under several graph operations and apply our results to find the F-index of different chemically interesting molecular graphs and nanostructures.

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Computing Fifth Geometric-Arithmetic Index for Circumcoronene series of benzenoid Hk

JOURNAL OF ADVANCES IN CHEMISTRY ◽

10.24297/jac.v3i1.915 ◽

2007 ◽

Vol 3 (1) ◽

pp. 143-148 ◽

Cited By ~ 2

Author(s):

Mohammad Reza Farahani

Keyword(s):

Connected Graph ◽

Topological Index ◽

Degree Of Vertex ◽

Simple Connected Graph ◽

Ga Index

Let G=(V; E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E (G), respectively. The geometric-arithmetic index is a topological index was introduced by Vukicevic and Furtula in 2009 and defined as in which degree of vertex u denoted by dG(u) (or du for short). In 2011, A. Graovac et al defined a new version of GA index as where The goal of this paper is to compute the fifth geometric-arithmetic index for "Circumcoronene series of benzenoid Hk (k≥1)".

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Reformulated Zagreb Indices of Some Derived Graphs

Mathematics ◽

10.3390/math7040366 ◽

2019 ◽

Vol 7 (4) ◽

pp. 366 ◽

Cited By ~ 6

Author(s):

Jia-Bao Liu ◽

Bahadur Ali ◽

Muhammad Aslam Malik ◽

Hafiz Muhammad Afzal Siddiqui ◽

Muhammad Imran

Keyword(s):

Physical Properties ◽

Chemical Structure ◽

Topological Index ◽

Chemical Reactivity ◽

Line Graph ◽

Total Graph ◽

Zagreb Indices ◽

Subdivision Graph ◽

Chemical Constitution ◽

Vertex Degrees

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph.

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Electron Energy Studying of Molecular Structures via Forgotten Topological Index Computation

Journal of Chemistry ◽

10.1155/2016/1053183 ◽

2016 ◽

Vol 2016 ◽

pp. 1-7 ◽

Cited By ~ 29

Author(s):

Wei Gao ◽

Weifan Wang ◽

Muhammad Kamran Jamil ◽

Mohammad Reza Farahani

Keyword(s):

Carbon Atom ◽

Electron Energy ◽

Topological Index ◽

Molecular Graph ◽

Chemical Properties ◽

Sum Of Squares ◽

Molecular Structures ◽

Physical Chemical ◽

Vertex Degrees ◽

Chemical Material

It is found from the earlier studies that the structure-dependency of totalπ-electron energyEπheavily relies on the sum of squares of the vertex degrees of the molecular graph. Hence, it provides a measure of the branching of the carbon-atom skeleton. In recent years, the sum of squares of the vertex degrees of the molecular graph has been defined as forgotten topological index which reflects the structure-dependency of totalπ-electron energyEπand measures the physical-chemical properties of molecular structures. In this paper, in order to research the structure-dependency of totalπ-electron energyEπ, we present the forgotten topological index of some important molecular structures from mathematical standpoint. The formulations we obtained here use the approach of edge set dividing, and the conclusions can be applied in physics, chemical, material, and pharmaceutical engineering.

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Zagreb, multiplicative zagreb indices and coindices of 〖nc〗_n (k) and 〖ca〗_3 (c_6 ) graphs

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v36i2.30613 ◽

2018 ◽

Vol 36 (2) ◽

pp. 9-15

Author(s):

Vida Ahmadi ◽

Mohammad Reza Darafshe

Keyword(s):

Connected Graph ◽

Zagreb Indices ◽

The Third ◽

Degree Of Vertex ◽

Vertex Set ◽

Simple Connected Graph

Let be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are defind, respectivly by: , and where is the degree of vertex u in G and uv is an edge of G, connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices of graph are defind by: and . The first and second Zagreb coindices of graph are defind by: and . and , named as multiplicative Zagreb coindices. In this article, we compute the first, second and the third Zagreb indices and the first and second multiplicative Zagreb indices of some graphs. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.

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Relations between ordinary and multiplicative degree-based topological indices

Filomat ◽

10.2298/fil1808031g ◽

2018 ◽

Vol 32 (8) ◽

pp. 3031-3042 ◽

Cited By ~ 1

Author(s):

Ivan Gutman ◽

Igor Milovanovic ◽

Emina Milovanovic

Keyword(s):

Lower Bounds ◽

Connected Graph ◽

Topological Indices ◽

Upper And Lower Bounds ◽

First Zagreb Index ◽

Zagreb Index ◽

Vertex Degrees ◽

Simple Connected Graph

Let G be a simple connected graph with n vertices and m edges, and sequence of vertex degrees d1 ? d2 ?...? dn > 0. If vertices i and j are adjacent, we write i ~ j. Denote by ?1, ?*1, Q? and H? the multiplicative Zagreb index, multiplicative sum Zagreb index, general first Zagreb index, and general sumconnectivity index, respectively. These indices are defined as ?1 = ?ni=1 d2i, ?*1 = ?i~j(di+dj), Q? = ?n,i=1 d?i and H? = ?i~j(di+dj)?. We establish upper and lower bounds for the differences H?-m (?1*)?/m and Q?-n(?1)?/2n . In this way we generalize a number of results that were earlier reported in the literature.

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Lower Bounds for Inverse Sum Indeg Index of Graphs

Kragujevac Journal of Mathematics ◽

10.46793/kgjmat2004.551g ◽

2020 ◽

Vol 44 (4) ◽

pp. 551-562

Author(s):

Ivan Gutman ◽

M. MATEJIC ◽

E. MILOVANOVIC ◽

I. MILOVANOVIC

Keyword(s):

Lower Bounds ◽

Connected Graph ◽

Vertex Degrees ◽

Simple Connected Graph

Let G = (V,E), V = {1, 2,…,n}, be a simple connected graph with n vertices and m edges and let d1 ≥ d2 ≥⋅ ⋅⋅≥ dn > 0, be the sequence of its vertex degrees. With i ∼ j we denote the adjacency of the vertices i and j in G. The inverse sum indeg index is deﬁned as ISI = ∑ -didj- di+dj with summation going over all pairs of adjacent vertices. We consider lower bounds for ISI. We ﬁrst analyze some lower bounds reported in the literature. Then we determine some new lower bounds.

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Chromatic Zagreb indices for graphical embodiment of colour clusters

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2019.3.1.6 ◽

2019 ◽

Vol 3 (1) ◽

pp. 48

Author(s):

Johan Kok ◽

Sudev Naduvath ◽

Muhammad Kamran Jamil

Keyword(s):

Positive Integer ◽

Connected Graph ◽

Chromatic Numbers ◽

Zagreb Indices ◽

Colour Class ◽

Graph Structures ◽

Simple Connected Graph

For a colour cluster C = (C1, C2, C3, …, Cℓ), where Ci is a colour class such that ∣Ci∣ = ri, a positive integer, we investigate two types of simple connected graph structures G1C, G2C which represent graphical embodiments of the colour cluster such that the chromatic numbers χ(G1C) = χ(G2C) = ℓ and $\min\{\varepsilon(G^{C}_1)\}=\min\{\varepsilon(G^{C}_2)\} =\sum\limits_{i=1}^{\ell}r_i-1$, and ɛ(G) is the size of a graph G. In this paper, we also discuss the chromatic Zagreb indices corresponding to G1C, G2C.

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