Inverse Sum Indeg Index (Energy) with Applications to Anticancer Drugs

For a simple graph with vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><msub><mi>v</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>v</mi><mi>n</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> with degree sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><msub><mi>v</mi><mi>i</mi></msub></msub></semantics></math></inline-formula> of vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>v</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula>, the inverse sum indeg matrix (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>I</mi></mrow></semantics></math></inline-formula>-matrix) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>I</mi><mi>S</mi><mi>I</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>(</mo><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></mrow></semantics></math></inline-formula> of <i>G</i> is defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>d</mi><msub><mi>v</mi><mi>i</mi></msub></msub><msub><mi>d</mi><msub><mi>v</mi><mi>j</mi></msub></msub></mrow><mrow><msub><mi>d</mi><msub><mi>v</mi><mi>i</mi></msub></msub><mo>+</mo><msub><mi>d</mi><msub><mi>v</mi><mi>j</mi></msub></msub></mrow></mfrac><mo>,</mo></mrow></semantics></math></inline-formula> if <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> is adjacent to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>j</mi></msub></semantics></math></inline-formula>, and zero, otherwise. The multiset of eigenvalues of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>I</mi><mi>S</mi><mi>I</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>I</mi></mrow></semantics></math></inline-formula>-spectrum of <i>G</i> and the sum of their absolute values is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>I</mi></mrow></semantics></math></inline-formula>-energy of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula>. In this paper, we modify the two results of (Li, Ye and Broersma, 2022), give the correct characterization of the extremal graphs and thereby obtain better bounds than the already known results. Moreover, we also discuss the QSPR analysis and carry the statistical modelling (linear, logarithmic and quadratic) of the physicochemical properties of anticancer drugs with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>I</mi></mrow></semantics></math></inline-formula>-index (energy).