scholarly journals Note on the Reformulated Zagreb Indices of Two Classes of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Tongkun Qu ◽  
Mengya He ◽  
Shengjin Ji ◽  
Xia Li
Keyword(s):  
Upper Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Vertex Degrees

The reformulated Zagreb indices of a graph are obtained from the original Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of its two end vertices minus 2. In this paper, we obtain two upper bounds of the first reformulated Zagreb index among all graphs with p pendant vertices and all graphs having key vertices for which they will become trees after deleting their one key vertex. Moreover, the corresponding extremal graphs which attained these bounds are characterized.

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 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 Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Wei Gao ◽  
Muhammad Kamran Jamil ◽  
Aisha Javed ◽  
Mohammad Reza Farahani ◽  
Shaohui Wang ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Graph Transformations ◽  
Mathematical Properties ◽  
Bicyclic Graphs ◽  
Important Branch

The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as∑uv∈E(G)‍(d(u)+d(v))2, whered(v)is the degree of the vertexvin a graphG=(V(G),E(G)). In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using these nice mathematical properties, the extremal graphs amongn-vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index. Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.

scholarly journals F-Index of some graph operations

2016 ◽  
Vol 08 (02) ◽  
pp. 1650025 ◽  
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.

SOME RESULTS ON THE DIFFERENCE OF THE ZAGREB INDICES OF A GRAPH

2015 ◽  
Vol 92 (2) ◽  
pp. 177-186 ◽  
Author(s):  
MINGQIANG AN ◽  
LIMING XIONG
Keyword(s):  
Independence Number ◽  
Matching Number ◽  
Extremal Graphs ◽  
Vertex Connectivity ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
The Difference ◽  
Appl Math

The classical first and second Zagreb indices of a graph $G$ are defined as $M_{1}(G)=\sum _{v\in V(G)}d(v)^{2}$ and $M_{2}(G)=\sum _{e=uv\in E(G)}d(u)d(v),$ where $d(v)$ is the degree of the vertex $v$ of $G.$ Recently, Furtula et al. [‘On difference of Zagreb indices’, Discrete Appl. Math.178 (2014), 83–88] studied the difference of $M_{1}$ and $M_{2},$ and showed that this difference is closely related to the vertex-degree-based invariant $RM_{2}(G)=\sum _{e=uv\in E(G)}[d(u)-1][d(v)-1]$, the reduced second Zagreb index. In this paper, we present sharp bounds for the reduced second Zagreb index, given the matching number, independence number and vertex connectivity, and we also completely determine the extremal graphs.

scholarly journals On the maximum and minimum first reformulated Zagreb index of graphs with connectivity at most k

Filomat ◽  
2011 ◽  
Vol 25 (4) ◽  
pp. 75-83 ◽  
Author(s):  
Guifu Su ◽  
Liming Xiong ◽  
Lan Xu ◽  
Beibei Ma
Keyword(s):  
Graph Theory ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Zagreb Indices ◽  
Chemical Graph Theory

The authors Milicevic et al. introduced the reformulated Zagreb indices [1], which is a generalization of classical Zagreb indices of chemical graph theory. In this paper, we mainly consider the maximum and minimum for the first reformulated index of graphs with connectivity at most k. The corresponding extremal graphs are characterized.

scholarly journals Computing Topological Indices and Polynomials for Line Graphs

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 137 ◽  
Author(s):  
Shahid Imran ◽  
Muhammad Siddiqui ◽  
Muhammad Imran ◽  
Muhammad Nadeem
Keyword(s):  
Topological Index ◽  
Line Graph ◽  
Topological Indices ◽  
Line Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Vertex Degrees ◽  
Set Up ◽  
Ladder Graphs

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision.

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 General Multiplicative Zagreb Indices of Graphs with a Small Number of Cycles

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 514 ◽  
Author(s):  
Monther R. Alfuraidan ◽  
Tomáš Vetrík ◽  
Selvaraj Balachandran
Keyword(s):  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Zagreb Indices ◽  
Special Cases ◽  
Tricyclic Graphs ◽  
Bicyclic Graphs ◽  
Number Of Cycles

We present lower and upper bounds on the general multiplicative Zagreb indices for bicyclic graphs of a given order and number of pendant vertices. Then, we generalize our methods and obtain bounds for the general multiplicative Zagreb indices of tricyclic graphs, tetracyclic graphs and graphs of given order, size and number of pendant vertices. We show that all our bounds are sharp by presenting extremal graphs including graphs with symmetries. Bounds for the classical multiplicative Zagreb indices are special cases of our results.

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