<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Many 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 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amns.2020.2.00020_ieq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mover accent="true"><mi>M</mi><mo>¯</mo></mover></mrow><mn>1</mn><mi>α</mi></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi mathvariant="italic">uv</mi><mo>∈</mo><mi>E</mi><mo stretchy="false">(</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mi>α</mi></msup><mo>+</mo><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><mi>α</mi></msup><mo stretchy="false">]</mo></mrow></math><tex-math>\overline{M}^\alpha_1(G)=\Sigma_{uv\in E(\overline{G})}[d_G(u)^\alpha+d_G(v)^\alpha]</tex-math></alternatives></inline-formula>
, where <italic>α</italic> ∈ ℝ, <italic>α</italic> ≠ 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.</p></abstract>

Many 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 
M¯1α(G)=∑uv∈E(G¯)[dG(u)α+dG(v)α]\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.

The first general Zagreb coindex of graph operations

Addis Ababa Science and Technology University

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"The first general Zagreb coindex of graph operations"}

Many chemically important graphs can be obtained from simpler graphs by applying different graph operations. Graph operations such as union, sum, Cartesian...

The first general Zagreb coindex of graph operations

Published Online: Aug 14, 2020

Page range: 109 - 120

Received: Oct 26, 2019

Accepted: Nov 18, 2019

DOI: https://doi.org/10.2478/amns.2020.2.00020

Keywords
Topological index, first general Zagreb index, first general Zagreb coindex, Graph operation

© 2020 Melaku Berhe Belay, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

The first general Zagreb coindex of graph operations

Published Online: Aug 14, 2020

Page range: 109 - 120

Received: Oct 26, 2019

Accepted: Nov 18, 2019

DOI: https://doi.org/10.2478/amns.2020.2.00020

KeywordsTopological index, first general Zagreb index, first general Zagreb coindex, Graph operation

© 2020 Melaku Berhe Belay, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Topological index, first general Zagreb index, first general Zagreb coindex, Graph operation