On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices

On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices

This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>...

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Bibliographic Details
Main Authors: Abeer M. Albalahi, Muhammad Rizwan, Akhlaq A. Bhatti, Ivan Gutman, Akbar Ali, Tariq Alraqad, Hicham Saber
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Axioms
Subjects:
bond incident degree indices
variable sum exdeg index
zeroth-order general Randić index
extremal values
tree
Online Access:https://www.mdpi.com/2075-1680/14/1/23
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Summary:This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and the general zeroth-order Randić index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula>. The minimum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> in the class of all trees of fixed order containing no vertex of even degree are obtained for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>; also, the maximum value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> in the mentioned class is determined for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> are found for every real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∉</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> are determined when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> does not belong to the closed interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>; in the same class, the maximum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> are also found for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The graphs that achieve the obtained extremal values are also determined.
ISSN:2075-1680

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