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 The Reciprocal Complementary Wiener Number of Graph Operations

2021 ◽  
Vol 45 (01) ◽  
pp. 139-154
Author(s):  
R. NASIRI ◽  
A. NAKHAEI ◽  
A. R. SHOJAEIFARD
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Symmetric Difference ◽  
Graph Operations ◽  
Strong Product ◽  
Product Composition ◽  
Wiener Number ◽  
Corona Product

The reciprocal complementary Wiener number of a connected graph G is defined as ∑ {x,y}⊆V (G) 1 D+1-−-dG(x,y), where D is the diameter of G and dG(x,y) is the distance between vertices x and y. In this work, we study the reciprocal complementary Wiener number of various graph operations such as join, Cartesian product, composition, strong product, disjunction, symmetric difference, corona product, splice and link of graphs.

scholarly journals HF-Index and Y-Index of Some New Graph of Operations

2022 ◽  
pp. 28-33
Author(s):  
Dr. S. Nagarajan ◽  
◽  
G. Kayalvizhi ◽  
G. Priyadharsini ◽  
◽  
...  
Keyword(s):  
Cartesian Product ◽  
Graph Operations ◽  
Join Of Graphs ◽  
Product Of Graphs ◽  
Corona Product

In this paper we derive HF index of some graph operations containing join, Cartesian Product, Corona Product of graphs and compute the Y index of new operations of graphs related to the join of graphs.

A novel graph invariant: The third leap Zagreb index under several graph operations

2019 ◽  
Vol 11 (05) ◽  
pp. 1950054 ◽  
Author(s):  
Durbar Maji ◽  
Ganesh Ghorai
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
The Third ◽  
Graph Operations ◽  
Strong Product ◽  
Corona Product

The third leap Zagreb index of a graph [Formula: see text] is denoted as [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are the 2-distance degree and the degree of the vertex [Formula: see text] in [Formula: see text], respectively. The first, second and third leap Zagreb indices were introduced by Naji et al. [A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Combin. Optim. 2(2) (2017) 99–117] in 2017. In this paper, the behavior of the third leap Zagreb index under several graph operations like the Cartesian product, Corona product, neighborhood Corona product, lexicographic product, strong product, tensor product, symmetric difference and disjunction of two graphs is studied.

General reduced second Zagreb index of graph operations

2020 ◽  
pp. 2150082
Author(s):  
R. Khoeilar ◽  
A. Jahanbani
Keyword(s):  
Real Number ◽  
Cartesian Product ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Graph Operations ◽  
Vertex Set ◽  
Join Of Graphs ◽  
Corona Product

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. The general reduced second Zagreb index of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is any real number and [Formula: see text] is the degree of the vertex [Formula: see text] of [Formula: see text]. In this paper, the general reduced second Zagreb index of the Cartesian product, corona product, join of graphs and two new operations of graphs are computed.

scholarly journals Sharp Bounds on the Augmented Zagreb Index of Graph Operations

2020 ◽  
Vol 44 (4) ◽  
pp. 509-522
Author(s):  
N. DEHGARDI ◽  
H. ARAM
Keyword(s):  
Simple Graph ◽  
Connectivity Index ◽  
Regular Graphs ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Graph Operations ◽  
Degree Of A Vertex ◽  
Augmented Zagreb Index ◽  
Composition Product ◽  
Product Of Graphs

Let G be a finite and simple graph with edge set E(G). The augmented Zagreb index of G is ( ) ∑ dG (u )dG (v) 3 AZI (G ) = ---------------------- , dG (u ) + dG (v) − 2 uv∈E(G ) where dG(u) denotes the degree of a vertex u in G. In this paper, we give some bounds of this index for join, corona, cartesian and composition product of graphs by general sum-connectivity index and general Randić index and compute the sharp amount of that for the regular graphs.

scholarly journals On the Reformulated Second Zagreb Index of Graph Operations

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Durbar Maji ◽  
Ganesh Ghorai ◽  
Yaé Ulrich Gaba
Keyword(s):  
Biological Activities ◽  
Cartesian Product ◽  
Real Numbers ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Graph Operations ◽  
Sum Of Products ◽  
Physical Attributes ◽  
Vertex Degrees ◽  
Corona Product

Topological indices (TIs) are expressed by constant real numbers that reveal the structure of the graphs in QSAR/QSPR investigation. The reformulated second Zagreb index (RSZI) is such a novel TI having good correlations with various physical attributes, chemical reactivities, or biological activities/properties. The RSZI is defined as the sum of products of edge degrees of the adjacent edges, where the edge degree of an edge is taken to be the sum of vertex degrees of two end vertices of that edge with minus 2. In this study, the behaviour of RSZI under graph operations containing Cartesian product, join, composition, and corona product of two graphs has been established. We have also applied these results to compute RSZI for some important classes of molecular graphs and nanostructures.

scholarly journals The first general Zagreb coindex of graph operations

2020 ◽  
Vol 5 (2) ◽  
pp. 109-120
Author(s):  
Melaku Berhe Belay ◽  
Chunxiang Wang
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Molecular Graphs ◽  
Basic Properties ◽  
Graph Operations ◽  
Nano Structures ◽  
Product Composition ◽  
Product Of Graphs

AbstractMany chemically important graphs can be obtained from simpler graphs by applying different graph operations. Graph operations such as union, sum, Cartesian product, composition and tensor product of graphs are among the important ones. In this paper, we introduce a new invariant which is named as the first general Zagreb coindex and defined as \overline{M}^\alpha_1(G)=\Sigma_{uv\in E(\overline{G})}[d_G(u)^\alpha+d_G(v)^\alpha] , where α ∈ ℝ, α ≠ 0. Here, we study the basic properties of the newly introduced invariant and its behavior under some graph operations such as union, sum, Cartesian product, composition and tensor product of graphs and hence apply the results to find the first general Zagreb coindex of different important nano-structures and molecular graphs.

scholarly journals Inverse sum indeg coindex of graphs

2019 ◽  
Vol 11 (2) ◽  
pp. 399-406
Author(s):  
K. Pattabiraman
Keyword(s):  
Connected Graph ◽  
Upper Bounds ◽  
Product Graph ◽  
Degree Of A Vertex ◽  
Simple Connected Graph ◽  
Corona Product

The inverse sum indeg coindex $\overline{ISI}(G)$ of a simple connected graph $G$ is defined as the sum of the terms $\frac{d_G(u)d_G(v)}{d_G(u)+d_G(v)}$ over all edges $uv$ not in $G,$ where $d_G(u)$ denotes the degree of a vertex $u$ of $G.$ In this paper, we present the upper bounds on inverse sum indeg coindex of edge corona product graph and Mycielskian graph. In addition, we obtain the exact value of both inverse sum indeg index and its coindex of a double graph.

scholarly journals Joins, coronas and their vertex-edge Wiener polynomials

2016 ◽  
Vol 47 (2) ◽  
pp. 163-178
Author(s):  
Mahdieh Azari ◽  
Ali Iranmanesh
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Wiener Index ◽  
First Derivative ◽  
Sum Of Distances ◽  
Simple Connected Graph ◽  
Product Of Graphs ◽  
Corona Product

The vertex-edge Wiener index of a simple connected graph $G$ is defined as the sum of distances between vertices and edges of $G$. The vertex-edge Wiener polynomial of $G$ is a generating function whose first derivative is a $q-$analog of the vertex-edge Wiener index. Two possible distances $D_1(u, e|G)$ and $D_2(u, e|G)$ between a vertex $u$ and an edge $e$ of $G$ can be considered and corresponding to them, the first and second vertex-edge Wiener indices of $G$, and the first and second vertex-edge Wiener polynomials of $G$ are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.

scholarly journals Proper Vertex Connection and Graph Operations

2019 ◽  
Vol 19 (02) ◽  
pp. 1950001
Author(s):  
YINGYING ZHANG ◽  
XIAOYU ZHU
Keyword(s):  
Direct Product ◽  
Connected Graph ◽  
Upper Bounds ◽  
Lexicographic Product ◽  
Colored Graph ◽  
Connection Number ◽  
Graph Operations ◽  
Proper Vertex

A path in a vertex-colored graph is a vertex-proper path if any two internal adjacent vertices differ in color. A vertex-colored graph is proper vertex k-connected if any two vertices of the graph are connected by k disjoint vertex-proper paths of the graph. For a k-connected graph G, the proper vertex k-connection number of G, denoted by pvck(G), is defined as the smallest number of colors required to make G proper vertex k-connected. A vertex-colored graph is strong proper vertex-connected, if for any two vertices u, v of the graph, there exists a vertex-proper u-v geodesic. For a connected graph G, the strong proper vertex-connection number of G, denoted by spvc(G), is the smallest number of colors required to make G strong proper vertex-connected. In this paper, we study the proper vertex k-connection number and the strong proper vertex-connection number on the join of two graphs, the Cartesian, lexicographic, strong and direct product, and present exact values or upper bounds for these operations of graphs. Then we apply these results to some instances of Cartesian and lexicographic product networks.

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