| Summary: | For a simple finite graph <i>G</i>, the generalized adjacency matrix is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi mathvariant="script">D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> are respectively the adjacency matrix and diagonal matrix of the vertex degrees. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-spread of a graph <i>G</i> is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-spread of a graph. <i>Electron. J. Linear Algebra </i><b>2020</b>, <i>36</i>, 214–227). Furthermore, we show that the path graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula>, has the smallest <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> among all trees of order <i>n</i>. We establish a relationship between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> We obtain several bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><msub><mi mathvariant="script">A</mi><mi>α</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>.
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