| Summary: | Let <i>G</i> be a graph of order <i>n</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> be its Laplacian matrix. The Laplacian polynomial of <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mrow><mo>(</mo><mi>G</mi><mo>;</mo><mi>λ</mi><mo>)</mo></mrow><mo>=</mo><mo movablelimits="true" form="prefix">det</mo><mrow><mo>(</mo><mi>λ</mi><mi>I</mi><mo>−</mo><mi>L</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></msubsup><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>i</mi></msup><msub><mi>c</mi><mi>i</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><msup><mi>λ</mi><mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is called the <i>i</i>-th Laplacian coefficient of <i>G</i>. Denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">G</mi><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></semantics></math></inline-formula> the set of all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula>-graphs, in which each of them contains <i>n</i> vertices and <i>m</i> edges. The graph <i>G</i> is called <i>uniformly minimal</i> if, for each <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>(</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <i>H</i> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-minimal in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">G</mi><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></semantics></math></inline-formula>. The Laplacian matrix and eigenvalues of graphs have numerous applications in various interdisciplinary fields, such as chemistry and physics. Specifically, these matrices and eigenvalues are widely utilized to calculate the energy of molecular energy and analyze the physical properties of materials. The Laplacian-like energy shares a number of properties with the usual graph energy. In this paper, we investigate the existence of uniformly minimal graphs in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">G</mi><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></semantics></math></inline-formula> because such graphs have minimal Laplacian-like energy. We determine that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>3</mn></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> successive minimal graph is exactly one of the four classes of threshold graphs.
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