On the Exponential Atom-Bond Connectivity Index of Graphs
Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential verte...
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| Format: | Article |
| Language: | English |
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MDPI AG
2025-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/2/269 |
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| Summary: | Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mi mathvariant="script">ABC</mi></msup><mo>=</mo><msup><mi>e</mi><mi mathvariant="script">ABC</mi></msup><mrow><mo>(</mo><mo>Υ</mo><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mo>Υ</mo><mo>)</mo></mrow></mrow></munder></mstyle><mspace width="0.166667em"></mspace><msup><mi>e</mi><msqrt><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msub><mi>d</mi><mi>i</mi></msub><mo>+</mo><msub><mi>d</mi><mi>j</mi></msub><mo>−</mo><mn>2</mn></mrow><mrow><msub><mi>d</mi><mi>i</mi></msub><mspace width="0.166667em"></mspace><msub><mi>d</mi><mi>j</mi></msub></mrow></mfrac></mstyle></msqrt></msup><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>i</mi></msub></semantics></math></inline-formula> is the degree of the vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Υ</mo></semantics></math></inline-formula>. In this paper, we prove that the double star <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><msub><mi>S</mi><mrow><mi>n</mi><mo>−</mo><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> is the second maximal graph with respect to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>e</mi><mi mathvariant="script">ABC</mi></msup></semantics></math></inline-formula> index of trees of order <i>n</i>. We give an upper bound on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>e</mi><mi mathvariant="script">ABC</mi></msup></semantics></math></inline-formula> of unicyclic graphs of order <i>n</i> and characterize the maximal graphs. The graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>∨</mo><mrow><mo>(</mo><msub><mi>P</mi><mn>3</mn></msub><mo>∪</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo>)</mo></mrow><msub><mi>K</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> gives the maximal graph with respect to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>e</mi><mi mathvariant="script">ABC</mi></msup></semantics></math></inline-formula> index of bicyclic graphs of order <i>n</i>. We present several relations between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mi mathvariant="script">ABC</mi></msup><mrow><mo>(</mo><mo>Υ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi><mo>(</mo><mo>Υ</mo><mo>)</mo></mrow></semantics></math></inline-formula> of graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Υ</mo></semantics></math></inline-formula>. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research. |
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| ISSN: | 2227-7390 |