Table 1 Common Fuzzy Graph Operations and Their Mathematical Definitions.

Fuzzy graph

¹⁰

G = (V, E, σ, μ) where:

V: a set of nodes

E ⊆ V × V: set of edges

σ: V → [0,1]: node membership function

μ: E → [0,1]: edge membership function

Basic fuzzy graph structure with node and edge membership values

Union

³

G1 + G2 = (V, σ, μ) where:

V = V1 ⋃ V2

σ(v) = σ1(v) if v \(\in\) V1

σ(v) = σ2(v) if v \(\in\) V2

μ(uv) = max{μ1(uv), μ2(uv)}

Combines two fuzzy graphs while preserving the highest membership values

Cartesian product⁷

G1 × G2 = (V, σ, μ) where:

V = V1 × V2

σ((u,v)) = min{σ1(u), σ2(v)}

μ((u1,v1)(u2,v2)) = min{μ1(u1u2), μ2(v1v2)}

Creates product graph with combined vertices and edges

Complement

⁶

G' = (V, σ, μ') where:

- μ'(uv) = min{σ(u), σ(v)}—μ(uv)

Inverts edge membership values while preserving vertices

Strong product

⁸

G1 \(\otimes\) G2 = (V, σ, μ) where:

V = V1 × V2

σ((u,v)) = σ1(u) ∧ σ2(v)

μ((u1,v1)(u2,v2)) = max{μ1(u1u2) ∧ σ2(v1), σ1(u1) ∧ μ2(v1v2)}

Combines graphs with stronger connectivity requirements