| Summary: | Let <i>G</i> be a graph on <i>n</i> vertices and <i>m</i> edges, with maximum degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and minimum degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> Let <i>A</i> be the adjacency matrix of <i>G</i>, and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mn>1</mn></msub><mo>≥</mo><msub><mi>λ</mi><mn>2</mn></msub><mo>≥</mo><mo>…</mo><mo>≥</mo><msub><mi>λ</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> be the eigenvalues of <i>G</i>. The energy of <i>G</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is defined as the sum of the absolute values of the eigenvalues of <i>G</i>, that is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo>|</mo></mrow><msub><mi>λ</mi><mn>1</mn></msub><mrow><mo>|</mo><mo>+</mo><mo>…</mo><mo>+</mo><mo>|</mo></mrow><msub><mi>λ</mi><mi>n</mi></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>. The energy of <i>G</i> is known to be at least twice the minimum degree of <i>G</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> Akbari and Hosseinzadeh conjectured that the energy of a graph <i>G</i> whose adjacency matrix is nonsingular is in fact greater than or equal to the sum of the maximum and the minimum degrees of <i>G</i>, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mo>Δ</mo><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, we present a proof of this conjecture for hyperenergetic graphs, and we prove an inequality that appears to support the conjectured inequality. Additionally, we derive various lower and upper bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. The results rely on elementary inequalities and their application.
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