scholarly journals Cactus Graphs with Maximal Multiplicative Sum Zagreb Index

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 913
Author(s):  
Chunlei Xu ◽  
Batmend Horoldagva ◽  
Lkhagva Buyantogtokh
Keyword(s):  
Connected Graph ◽  
Zagreb Index ◽  
Graph Transformations ◽  
Cactus Graphs ◽  
The Given ◽  
Extremal Properties

A connected graph G is said to be a cactus if any two cycles have at most one vertex in common. The multiplicative sum Zagreb index of a graph G is the product of the sum of the degrees of adjacent vertices in G. In this paper, we introduce several graph transformations that are useful tools for the study of the extremal properties of the multiplicative sum Zagreb index. Using these transformations and symmetric structural representations of some cactus graphs, we determine the graphs having maximal multiplicative sum Zagreb index for cactus graphs with the prescribed number of pendant vertices (cut edges). Furthermore, the graphs with maximal multiplicative sum Zagreb index are characterized among all cactus graphs of the given order.

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.

Edge (m,k)-choosability of graphs

2021 ◽  
Author(s):  
P. Soorya ◽  
K. A. Germina
Keyword(s):  
Connected Graph ◽  
Matching Number ◽  
Graph Theoretic ◽  
Maximum Value ◽  
Choice Number ◽  
The Given ◽  
Simple Connected Graph

Let [Formula: see text] be a simple, connected graph of order [Formula: see text] and size [Formula: see text] Then, [Formula: see text] is said to be edge [Formula: see text]-choosable, if there exists a collection of subsets of the edge set, [Formula: see text] of cardinality [Formula: see text] such that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are incident. This paper initiates a study on edge [Formula: see text]-choosability of certain fundamental classes of graphs and determines the maximum value of [Formula: see text] for which the given graph [Formula: see text] is edge [Formula: see text]-choosable. Also, in this paper, the relation between edge choice number and other graph theoretic parameters is discussed and we have given a conjecture on the relation between edge choice number and matching number of a graph.

scholarly journals Upper Bounds for the Modified Second Multiplicative Zagreb Index of Graph Operations

2016 ◽  
Vol 17 ◽  
pp. 10-16
Author(s):  
Bommanahal Basavanagoud ◽  
Shreekant Patil
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Upper Bounds ◽  
Zagreb Index ◽  
Graph Operations ◽  
Product Composition ◽  
Degree Of A Vertex ◽  
Product Of Graphs ◽  
Corona Product

The modified second multiplicative Zagreb index of a connected graph G, denoted by $\prod_{2}^{*}(G)$, is defined as $\prod_{2}^{*}(G)=\prod \limits_{uv\in E(G)}[d_{G}(u)+d_{G}(v)]^{[d_{G}(u)+d_{G}(v)]}$ where $d_{G}(z)$ is the degree of a vertex z in G. In this paper, we present some upper bounds for the modified second multiplicative Zagreb index of graph operations such as union, join, Cartesian product, composition and corona product of graphs are derived.The modified second multiplicative Zagreb index of aconnected graph , denoted by , is defined as where is the degree of avertex in . In this paper, we present some upper bounds for themodified second multiplicative Zagreb index of graph operations such as union,join, Cartesian product, composition and corona product of graphs are derived.

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 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 On the Aα-Spectral Radii of Cactus Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 869
Author(s):  
Chunxiang Wang ◽  
Shaohui Wang ◽  
Jia-Bao Liu ◽  
Bing Wei
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Common Vertex ◽  
Extremal Graphs ◽  
Spectral Matrix ◽  
Cactus Graph ◽  
Signless Laplacian ◽  
Cactus Graphs

Let A ( G ) be the adjacent matrix and D ( G ) the diagonal matrix of the degrees of a graph G, respectively. For 0 ≤ α ≤ 1 , the A α -matrix is the general adjacency and signless Laplacian spectral matrix having the form of A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Clearly, A 0 ( G ) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A α -spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results.

scholarly journals DEGREE DISTANCE AND MINIMUM DEGREE

2012 ◽  
Vol 87 (2) ◽  
pp. 255-271 ◽  
Author(s):  
S. MUKWEMBI ◽  
S. MUNYIRA
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Minimum Degree ◽  
Degree Of Vertex ◽  
Degree Distance ◽  
Image Position ◽  
Appl Math ◽  
Extremal Properties

AbstractLet G be a finite connected graph of order n, minimum degree δ and diameter d. The degree distance D′(G) of G is defined as ∑ {u,v}⊆V (G)(deg u+deg v) d(u,v), where deg w is the degree of vertex w and d(u,v) denotes the distance between u and v. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that \[ D^\prime (G)\le \frac {1}{4}\,dn\biggl (n-\frac {d}{3}(\delta +1)\biggr )^2+O(n^3). \] As a corollary, we obtain the bound D′ (G)≤n4 /(9(δ+1) )+O(n3) for a graph G of order n and minimum degree δ. This result, apart from improving on a result of Dankelmann et al. [‘On the degree distance of a graph’, Discrete Appl. Math.157 (2009), 2773–2777] for graphs of given order and minimum degree, completely settles a conjecture of Tomescu [‘Some extremal properties of the degree distance of a graph’, Discrete Appl. Math.98(1999), 159–163].

scholarly journals Neighbourhood Degree Based Index for Graph Operations Related to Composition

2019 ◽  
Vol 8 (4) ◽  
pp. 8723-8728
Keyword(s):  
Connected Graph ◽  
Zagreb Index ◽  
Connected Graphs ◽  
Graph Operations ◽  
Graph Operation ◽  
Simple Connected Graph

In this paper, we have investigated Zagreb index, F-index and neighbourhood degree based index for composition of two connected graphs in which one graph is obtained by using a new graph operation and other is a simple connected graph.

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.

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