On Sombor Index

• Main Authors: Kinkar Chandra Das, Ahmet Sinan Çevik, Ismail Naci Cangul, Yilun Shang
• Format: Article
• Language: English
• Published: MDPI AG 2021-01-01
• Series: Symmetry
• Subjects: graph; sombor index; maximum degree; minimum degree; independence number
• Online Access: https://www.mdpi.com/2073-8994/13/1/140

The concept of Sombor index <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>S</mi><mi>O</mi><mo>)</mo></mrow></semantics></math></inline-formula> was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mi>O</mi></mrow></semantics></math></inline-formula>: <inline-formula><math 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><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><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><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math 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 vertex <inline-formula><math display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula> in <i>G</i>. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work.