| Summary: | Let <i>G</i> be a graph with set of vertices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>=</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> and edge set <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>. Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>=</mo><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder></mstyle><mspace width="0.166667em"></mspace><msqrt><mrow><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>j</mi></msub><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>G</mi></msub><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the degree of the vertex <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> in <i>G</i>. In this paper we present some lower and upper bounds on the Sombor index of graph <i>G</i> in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs.
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