| Summary: | In this work, we investigate essential definitions, defining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula> as a simple graph with vertices in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></semantics></math></inline-formula> and subgraphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>u</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></semantics></math></inline-formula> as unit residue and quadratic residue graphs modulo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow></semantics></math></inline-formula>, respectively. The investigation extends to the degree of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>u</mi></mrow></msub></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>Γ</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></semantics></math></inline-formula>, illuminating the properties of these subgraphs in the context of quadratic congruences.
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