Extremal problems on exponential vertex-degree-based topological indices.

Area of Science:

• Graph theory
• Mathematical chemistry
• Combinatorial chemistry

Background:

• Topological indices are crucial in mathematical chemistry for predicting molecular properties.
• Existing research focuses on various graph indices, but optimal bounds and exponential forms require further investigation.

Purpose of the Study:

• To derive new lower and upper optimal bounds for general exponential graph indices.
• To establish novel inequalities involving generalized atom-bound connectivity ($ABC_\alpha$) and generalized second Zagreb ($M_2^\alpha$) indices.
• To address extremal problems related to the exponential forms of these topological indices ($e^{ABC_\alpha}$ and $e^{M_2^{\alpha}}$).

Main Methods:

• Utilizing graph theory principles to define and analyze graph indices.
• Applying mathematical inequalities to establish relationships between different topological indices.
• Solving optimization problems to determine extremal values for exponential graph indices.

Main Results:

• New optimal lower and upper bounds for general exponential graph indices have been obtained.
• Novel inequalities involving $ABC_\alpha$ and $M_2^\alpha$ are presented.
• Extremal problems for $e^{ABC_\alpha}$ and $e^{M_2^{\alpha}}$ have been successfully solved.

Conclusions:

• The findings contribute to a deeper understanding of graph indices and their exponential counterparts.
• The established bounds and inequalities offer valuable tools for further research in mathematical chemistry and graph theory.
• This work provides a foundation for exploring more complex graph structures and their associated properties.