Sharp Bounds and Extremal Fuzzy Graphs for the Fuzzy Sombor Index

Technical Summary: Sharp Bounds and Extremal Fuzzy Graphs for the Fuzzy Sombor Index

Problem Statement
The paper addresses the lack of systematic characterization regarding the extremal behavior of the Fuzzy Sombor Index ($SO_\mu$) within the framework of fuzzy graph theory. While the classical Sombor index has been extensively studied on crisp graphs—where paths typically minimize and stars maximize the index among trees—the extension to fuzzy graphs introduces new complexities. In fuzzy graphs, vertex degrees and edge strengths are continuous values in $(0, 1]$, and the index depends on both edge membership values $\mu(u, v)$ and fuzzy vertex degrees $\mu_u$. The study seeks to determine which fuzzy graph structures minimize or maximize $SO_\mu$ (and its generalization $SO_\mu^\alpha$ for $\alpha > 1$) under constraints of fixed order ($n$) and fixed fuzzy size ($m_\mu$). Additionally, the paper aims to establish sharp inequalities relating $SO_\mu$ to other established fuzzy topological indices, specifically the Fuzzy Zagreb, Randić, and Nirmala indices.

Methodology
The research employs analytical methods rooted in graph theory and optimization principles. The authors define the Fuzzy Sombor Index as:
$$SO_\mu(\Gamma) = \sum_{uv \in E(\Gamma)} \mu(u, v) \sqrt{\mu_u^2 + \mu_v^2}$$
where $\mu_u = \sum_{v \in V} \mu(u, v)$ is the fuzzy degree of vertex $u$. The methodology involves:

1. Extremal Analysis: Investigating specific families of fuzzy graphs (paths, stars, cycles, complete graphs, and unicyclic graphs) to identify structural configurations that yield minimum or maximum index values.
2. Convexity and Inequality Arguments: Utilizing the convexity of the function $f(\eta) = \eta \sqrt{x^2 + y^2}$ and properties of convex transformations (for $\alpha > 1$) to prove that concentrating edge memberships on specific topologies (like stars) maximizes the index, while dispersing them (like in paths) minimizes it.
3. Comparative Analysis: Deriving algebraic bounds that relate $SO_\mu$ to the Fuzzy First Zagreb Index ($M_1$), Fuzzy Randić Index ($R$), and Fuzzy Nirmala Index ($N$), particularly for regular and irregular fuzzy graphs.
4. Numerical Verification: Providing tabulated data for specific values of $n$ and $m_\mu$ to illustrate the behavior of the index across different graph families.

Key Contributions and Results
The paper establishes several definitive results regarding the bounds and extremal graphs for the Fuzzy Sombor Index:

• General Bounds:

  • Minimum: For connected fuzzy graphs with fixed order $n$ and fuzzy size $m_\mu$, the index is minimized by the fuzzy path ($P_n^f$) where edge memberships are distributed as uniformly as possible to balance fuzzy degrees. The absolute minimum (zero) is achieved by the empty fuzzy graph.
  • Maximum: The index is maximized by the uniform fuzzy complete graph ($K_n^f$) where all edge memberships are equal. For a fixed $m_\mu$, the maximum is achieved when memberships are concentrated on the edges of a complete graph.

• Extremal Trees and Unicyclic Graphs:

  • Among fuzzy trees with fixed $n$ and $m_\mu$, the fuzzy path ($P_n^f$) minimizes the index, and the fuzzy star ($S_n^f$) maximizes it. The star maximizes the index by concentrating the total fuzzy size on a single central vertex, creating a high-degree hub.
  • Among fuzzy unicyclic graphs, the index is minimized by the fuzzy cycle ($C_n^f$) with uniform edge memberships and maximized by a structure where the underlying graph is a cycle with a star attached (concentrating degrees).

• Generalized Index ($SO_\mu^\alpha$):

  • For $\alpha > 1$, the extremal graphs remain the same as for $\alpha = 1$. The fuzzy complete graph uniquely maximizes the generalized index, and the fuzzy path minimizes it, due to the strictly increasing and convex nature of the power function.

• Inequalities with Other Indices:

  • The paper derives sharp inequalities linking $SO_\mu$ to the Fuzzy Randić Index ($R$), showing $SO_\mu(\Gamma) \geq \sqrt{2} R(\Gamma) \cdot \delta_\mu$ (where $\delta_\mu$ is the minimum fuzzy degree).
  • Bounds are established relating $SO_\mu$ to the Fuzzy Zagreb Index ($M_1$) and the Fuzzy Nirmala Index ($N$), demonstrating that $SO_\mu$ occupies an intermediate position between linear and quadratic measures of vertex degrees.

Significance
The paper claims that these findings advance the theory of fuzzy graph indices by extending classical extremal results to the fuzzy domain. The authors highlight that while the underlying topological structures (paths, stars, cycles) that extremize the index in crisp graphs remain valid in the fuzzy context, the continuous nature of edge memberships allows for finer optimization. Specifically, the results suggest that the Fuzzy Sombor Index is a sensitive descriptor for detecting structural centralization and vulnerability in fuzzy networks. The ability to model "structural intensity" and degree variance makes the index a valuable tool for analyzing networks where uncertainty and varying connection strengths are present. The study provides a foundational framework for future research into fuzzy topological indices, offering precise bounds that can be used for network performance assessment and comparison.