scholarly journals On the first Zagreb index and multiplicative Zagreb coindices of graphs

2016 ◽  
Vol 24 (1) ◽  
pp. 153-176 ◽  
Author(s):  
Kinkar Ch. Das ◽  
Nihat Akgunes ◽  
Muge Togan ◽  
Aysun Yurttas ◽  
I. Naci Cangul ◽  
...  
Keyword(s):  
Degree Sequence ◽  
Molecular Graph ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Irregularity Index ◽  
Vertex Set ◽  
The First Zagreb Index

AbstractFor a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as, where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariantsandnamed as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) =. The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

scholarly journals Bounds on General Randić Index for F-Sum Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Xu Li ◽  
Maqsood Ahmad ◽  
Muhammad Javaid ◽  
Muhammad Saeed ◽  
Jia-Bao Liu
Keyword(s):  
Molecular Graph ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Randic Index ◽  
Structure Activity ◽  
General Randić Index ◽  
General Randic Index ◽  
The First Zagreb Index

A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire π-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.

On Energy and Laplacian Energy of Graphs

2016 ◽  
Vol 31 ◽  
pp. 167-186 ◽  
Author(s):  
Kinkar Das ◽  
Seyed Ahmad Mojalal
Keyword(s):  
Maximum Degree ◽  
Upper Bounds ◽  
Simple Graph ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
The Absolute ◽  
The First Zagreb Index

Let $G=(V,E)$ be a simple graph of order $n$ with $m$ edges. The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. The Laplacian energy of the graph $G$ is defined as \[ LE = LE(G)=\sum^n_{i=1}\left|\mu_i-\frac{2m}{n}\right| \] where $\mu_1,\,\mu_2,\,\ldots,\,\mu_{n-1},\,\mu_n=0$ are the Laplacian eigenvalues of graph $G$. In this paper, some lower and upper bounds for $\mathcal{E}(G)$ are presented in terms of number of vertices, number of edges, maximum degree and the first Zagreb index, etc. Moreover, a relation between energy and Laplacian energy of graphs is given.

scholarly journals Hyper Zagreb indices of composite graphs

2016 ◽  
Vol 4 (2) ◽  
pp. 47 ◽  
Author(s):  
Sharmila Devi ◽  
V. Kaladevi
Keyword(s):  
Molecular Graph ◽  
Sum Of Squares ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
The First Zagreb Index ◽  
Thorn Graph

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the degrees of vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. Similarly, the hyper Zagreb index is defined as the sum of square of degree of vertices over all the edges.  In this paper, First we obtain the hyper Zagreb indices of some derived graphs and the generalized transformations graphs. Finally, the hyper Zagreb indices of double, extended double, thorn graph, subdivision vertex corona of graphs, Splice and link graphs are obtained.

Inverse Sum Indeg Index of Subdivision, t-Subdivision Graphs, and Related Sums

2020 ◽  
pp. 104-119 ◽  
Author(s):  
Amitav Doley ◽  
Jibonjyoti Buragohain ◽  
A. Bharali
Keyword(s):  
Surface Area ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Total Surface Area ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Octane Isomers ◽  
Graph Parameters ◽  
The First Zagreb Index

The inverse sum indeg (ISI) index of a graph G is defined as the sum of the weights dG(u)dG(v)/dG(u)+dG(v) of all edges uv in G, where dG(u) is the degree of the vertex u in G. This index is found to be a significant predictor of total surface area of octane isomers. In this chapter, the authors present some lower and upper bounds for ISI index of subdivision graphs, t-subdivision graphs, s-sum and st -sum of graphs in terms of some graph parameters such as order, size, maximum degree, minimum degree, and the first Zagreb index. The extremal graphs are also characterized for their sharpness.

scholarly journals On the Randić Index of Corona Product Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Ismael G. Yero ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Degree Sequence ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Regular Graphs ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Randic Index ◽  
Product Graphs ◽  
Vertex Set ◽  
Corona Product

Let G be a graph with vertex set V=(v1,v2,…,vn). Let δ(vi) be the degree of the vertex vi∈V. If the vertices vi1,vi2,…,vih+1 form a path of length h≥1 in the graph G, then the hth order Randić index Rh of G is defined as the sum of the terms 1/δ(vi1)δ(vi2)⋯δ(vih+1) over all paths of length h contained (as subgraphs) in G. Lower and upper bounds for Rh, in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.

On Atom-Bond Connectivity Index

2011 ◽  
Vol 66 (1-2) ◽  
pp. 61-66 ◽  
Author(s):  
Bo Zhou ◽  
Rundan Xing
Keyword(s):  
Upper Bounds ◽  
Graph Invariant ◽  
Excellent Correlation ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
The Third ◽  
Connectivity Indices ◽  
Abc Index ◽  
The First Zagreb Index ◽  
Atom Bond

The atom-bond connectivity (ABC) index, introduced by Estrada et al. in 1998, displays an excellent correlation with the formation heat of alkanes. We give upper bounds for this graph invariant using the number of vertices, the number of edges, the Randi´c connectivity indices, and the first Zagreb index. We determine the unique tree with the maximum ABC index among trees with given numbers of vertices and pendant vertices, and the n-vertex trees with the maximum, and the second, the third, and the fourth maximum ABC indices for n ≥ 6.

scholarly journals Sharp bounds for the modified multiplicative Zagreb indices of graphs with vertex connectivity at most k

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4673-4685
Author(s):  
Haiying Wang ◽  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Molecular Graph ◽  
Upper Bounds ◽  
Two Dimensional ◽  
Quantitative Structure ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Vertex Connectivity ◽  
Zagreb Indices ◽  
Molecular Chirality ◽  
Structure Activity

Zagreb indices and their modified versions of a molecular graph originate from many practical problems such as two dimensional quantitative structure-activity (2D QSAR) and molecular chirality. Nowadays, they have become important invariants which can be used to characterize the properties of graphs from different aspects. LetVk n (or Ek n respectively) be a set of graphs of n vertices with vertex connectivity (or edge connectivity respectively) at most k. In this paper, we explore some properties of the modified first and second multiplicative Zagreb indices of graphs in Vkn and Ekn. By using analytic and combinatorial tools, we obtain some sharp lower and upper bounds for these indices of graphs in Vk n and Ekn. In addition, the corresponding extremal graphs which attain the lower or upper bounds are characterized. Our results enrich outcomes on studying Zagreb indices and the methods developed in this paper may provide some new tools for investigating the values on modified multiplicative Zagreb indices of other classes of graphs.

scholarly journals Sharp bounds on Zagreb indices of cacti with k pendant vertices

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1189-1200 ◽  
Author(s):  
Shuchao Li ◽  
Huangxu Yang ◽  
Qin Zhao
Keyword(s):  
Perfect Matching ◽  
Molecular Graph ◽  
Common Vertex ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Sum Of Products ◽  
Vertex Degrees ◽  
The First Zagreb Index

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of its vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. A connected graph G is a cactus if any two of its cycles have at most one common vertex. In this paper, we investigate the first and the second Zagreb indices of cacti with k pendant vertices. We determine sharp bounds for M1 -, M2 -values of n-vertex cacti with k pendant vertices. As a consequence, we determine the n-vertex cacti with maximal Zagreb indices and we also determine the cactus with a perfect matching having maximal Zagreb indices.

scholarly journals Computing First Zagreb index and F-index of New C-products of Graphs

2017 ◽  
Vol 2 (1) ◽  
pp. 285-298 ◽  
Author(s):  
B. Basavanagoud ◽  
Wei Gao ◽  
Shreekant Patil ◽  
Veena R. Desai ◽  
Keerthi G. Mirajkar ◽  
...  
Keyword(s):  
Molecular Graph ◽  
Sum Of Squares ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
The First Zagreb Index

AbstractFor a (molecular) graph, the first Zagreb index is equal to the sum of squares of the degrees of vertices, and the F-index is equal to the sum of cubes of the degrees of vertices. In this paper, we introduce sixty four new operations on graphs and study the first Zagreb index and F-index of the resulting graphs.

scholarly journals Energy of Nonsingular Graphs: Improving Lower Bounds

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Hajar Shooshtari ◽  
Jonnathan Rodriguez ◽  
Akbar Jahanbani ◽  
Abbas Shokri
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Adjacency Matrix ◽  
Degree Sequence ◽  
Simple Graph ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
The First Zagreb Index

Let G be a simple graph of order n and A be its adjacency matrix. Let λ 1 ≥ λ 2 ≥ … ≥ λ n be eigenvalues of matrix A . Then, the energy of a graph G is defined as ε G = ∑ i = 1 n λ i . In this paper, we will discuss the new lower bounds for the energy of nonsingular graphs in terms of degree sequence, 2-sequence, the first Zagreb index, and chromatic number. Moreover, we improve some previous well-known bounds for connected nonsingular graphs.

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