| Summary: | This paper is concerned with the general atom-bond sum-connectivity index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><msub><mi>S</mi><mi>γ</mi></msub></mrow></semantics></math></inline-formula>, which is a generalization of the recently proposed atom-bond sum-connectivity index, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> is any real number. For a connected graph <i>G</i> with more than two vertices, the number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><msub><mi>S</mi><mi>γ</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined as the sum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><msup><mrow><mo>(</mo><msub><mi>d</mi><mi>x</mi></msub><mo>+</mo><msub><mi>d</mi><mi>y</mi></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mi>γ</mi></msup></semantics></math></inline-formula> over all edges <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>y</mi></mrow></semantics></math></inline-formula> of the graph <i>G</i>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>x</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>y</mi></msub></semantics></math></inline-formula> represent the degrees of the vertices <i>x</i> and <i>y</i> of <i>G</i>, respectively. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mn>10</mn><mo>≤</mo><mi>γ</mi><mo>≤</mo><mn>10</mn></mrow></semantics></math></inline-formula>, the significance of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><msub><mi>S</mi><mi>γ</mi></msub></mrow></semantics></math></inline-formula> is examined on the data set of twenty-five benzenoid hydrocarbons for predicting their enthalpy of formation. It is found that the predictive ability of the index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><msub><mi>S</mi><mi>γ</mi></msub></mrow></semantics></math></inline-formula> for the selected property of the considered hydrocarbons is comparable to other existing general indices of this type. The effect of the addition of an edge between two non-adjacent vertices of a graph under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><msub><mi>S</mi><mi>γ</mi></msub></mrow></semantics></math></inline-formula> is also investigated. Furthermore, several extremal results regarding trees, general graphs, and triangle-free graphs of a given number of vertices are proved.
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