scholarly journals Relations between ordinary and multiplicative degree-based topological indices

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 3031-3042 ◽  
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.

scholarly journals Lower and Upper Bounds for Hyper-Zagreb Index of Graphs

2020 ◽  
pp. 1401-1406
Author(s):  
G. H. SHIRDEL ◽  
H. REZAPOUR ◽  
R. NASIRI
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Chemical Compounds ◽  
Upper Bounds ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Simple Connected Graph

The topological indices are functions on the graph that do not depend on the labeling of their vertices. They are used by chemists for studying the properties of chemical compounds.  Let  be a simple connected graph. The Hyper-Zagreb index of the graph ,  is defined as  ,where  and  are the degrees of vertex  and , respectively. In this paper, we study the Hyper-Zagreb index and give upper and lower bounds for .

scholarly journals First and Second Zagreb Polynomials of VC5C7[p,q] and HC5C7[p,q]Nanotubes

2014 ◽  
Vol 31 ◽  
pp. 56-62 ◽  
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Topological Indices ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Connectivity Indices ◽  
Anti Inflammatory ◽  
Simple Connected Graph

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. There exist many topological indices and connectivity indices in graph theory. The First and Second Zagreb indices were first introduced by Gutman and Trinajstić In1972. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. In this paper, we focus on the structure of ”G = VC5C7[p,q]”and ”H = HC5C7[p,q]” nanotubes and counting first Zagreb index Zg1(G) = ∑veVdv2 and Second Zagreb index Zg2(G) =∑e=uveE(G)(du·dv) of G and H, as well as First Zagreb polynomial Zg1(G,x ) =∑e=uveE(G)xdu+dv and Second Zagreb Polynomial Zg2(G,x) = ∑e=uveE(G)xdu·dv

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

scholarly journals Bounds for symmetric division deg index of graphs

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 683-698 ◽  
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.

scholarly journals Computation of General Zagreb Index of Nanotubes Covered by C5 and C7

2020 ◽  
Vol 11 (1) ◽  
pp. 8001-8008
Keyword(s):  
Carbon Nanotubes ◽  
Connected Graph ◽  
Molecular Graph ◽  
Topological Indices ◽  
Topological Descriptors ◽  
Chemical Bonds ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Physical And Chemical ◽  
Simple Connected Graph

A molecular graph is hydrogen deleted simple connected graph in which vertices and edges are represented by atoms and chemical bonds, respectively. Topological indices are numerical parameters of a molecular graph which characterize its topology and are usually graph invariant. In Mathematical chemistry, topological descriptors play an important role in modeling different physical and chemical activities of molecules. In this study, the generalized Zagreb index for three types of carbon nanotubes is computed. By putting some particular values to the parameters, some important degree-based topological indices are also derived.

A Novel Weighted First Zagreb Index of Graph

2020 ◽  
pp. 92-103
Author(s):  
Jibonjyoti Buragohain ◽  
A. Bharali
Keyword(s):  
Connected Graph ◽  
Chemical Properties ◽  
Topological Indices ◽  
First Zagreb Index ◽  
Acentric Factor ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Physico Chemical ◽  
Octane Isomers ◽  
The First Zagreb Index

The Zagreb indices are the oldest among all degree-based topological indices. For a connected graph G, the first Zagreb index M1(G) is the sum of the term dG(u)+dG(v) corresponding to each edge uv in G, that is, M1 , where dG(u) is degree of the vertex u in G. In this chapter, the authors propose a weighted first Zagreb index and calculate its values for some standard graphs. Also, the authors study its correlations with various physico-chemical properties of octane isomers. It is found that this novel index has strong correlation with acentric factor and entropy of octane isomers as compared to other existing topological indices.

scholarly journals More on the Laplacian Estrada index

2009 ◽  
Vol 3 (2) ◽  
pp. 371-378 ◽  
Author(s):  
Bo Zhou ◽  
Ivan Gutman
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Estrada Index ◽  
The First Zagreb Index

Let G be a graph with n vertices and let ?1, ?2, . . . , ?n be its Laplacian eigenvalues. In some recent works a quantity called Laplacian Estrada index was considered, defined as LEE(G)?n1 e?i. We now establish some further properties of LEE, mainly upper and lower bounds in terms of the number of vertices, number of edges, and the first Zagreb index.

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 Lower Bounds for Inverse Sum Indeg Index of Graphs

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.

scholarly journals Sharp bounds on the signless Laplacian Estrada index of graphs

Filomat ◽  
2014 ◽  
Vol 28 (10) ◽  
pp. 1983-1988 ◽  
Author(s):  
Shan Gao ◽  
Huiqing Liu
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Estrada Index ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
The First Zagreb Index

Let G be a connected graph with n vertices and m edges. Let q1, q2,..., qn be the eigenvalues of the signless Laplacian matrix of G, where q1 ? q2 ? ... ? qn. The signless Laplacian Estrada index of G is defined as SLEE(G) = nPi=1 eqi. In this paper, we present some sharp lower bounds for SLEE(G) in terms of the k-degree and the first Zagreb index, respectively.

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