<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

Data publikacji: 14 sie 2020

Zakres stron: 109 - 120

Otrzymano: 26 paź 2019

Przyjęty: 18 lis 2019

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

Słowa kluczowe
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

Data publikacji: 14 sie 2020

Zakres stron: 109 - 120

Otrzymano: 26 paź 2019

Przyjęty: 18 lis 2019

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

Słowa kluczoweTopological 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.

Słowa kluczowe
Topological index, first general Zagreb index, first general Zagreb coindex, Graph operation