¹⁶: A fuzzy graph G=(V,σ,μ) is a triple consisting of a non-empty set V (a set of vertices) together with a pair of functions: σ:V→[0,1] , a mapping that gives a membership degree for every vertex v∈V and μ:V×V→[0,1] is a fuzzy relation that captures the connection strength for every vertex pair, such that μ(u,v)≤min{σ(u),σ(v)}∀(u,v)∈(u,v)∈V×V.

³⁵: The strength of path P is the weight of the weakest edge. That means S(P)=min{μ(u1,ui+1)} .

³⁵: Let G=(V,σ,μ) be a fuzzy graph, and uandv are vertices in G. The strength of connectedness between uandv is the maximum strength of all paths b/n uandv . Denoted by CONNG(u,v)orμ∞(u,v) .

^20,21: Let G = (V, σ, μ) be a fuzzy graph. Then, G is a fuzzy tree if G has a Fuzzy spanning subgraph, F=(V,σ,ν) , which is a tree, and for all (a,b)∈μ−ν , μ(a,b)<ν∞(a,b) .

³⁶: Let G=(A,B) be an IFG and x,y∈V⊆G . Then,

The ⱷ2− strength of connectedness between 𝑥 and 𝑦 is ⱷ2∞(x,y)=sup{ⱷ2k(x,y)/k=1,2,….n} where, ⱷ2k(x,y)=min{ⱷ2(x,v1),ⱷ2(v1,v2),………..,ⱷ2(vn−1,y)} .

The γ2− strength of connectedness between 𝑥 and 𝑦 is.

γ2∞((x,y)=Inf{γ2k(x,y)/k=1,2,….n}

where ,γ2k(x,y)=max{γ2(x,v1),γ2(v1,v2),………..,γ2(vn−1,y)} .

³⁷: Let G=(A,B) be a connected IFG. Then

²³: A neutrosophic graph (NG) on X(≠ϕ) is a pair G~=(ϑ,ξ ) where ϑ:X→]0−,1+[ is Neutrosophic set in X and ξ:V×V→]0−,1+[ is Neutrosophic relation on X such that ∀u, x ∈ X :

ξt(ux)≤ϑt(u)∧ϑt(x),ξi(ux)≤ϑi(u)∧ϑi(x)andξf(ux)≤ϑf(u)∨ϑf(x)with0−≤ξt(ux)+ξi(ux)+ξf(ux)≤3+

ϑ and ξ are neutrosophic vertex and edge set of G~ , respectively.

²³: Let G~=(ϑ,ξ ) be a SVNG. The T, I and F - strength of connectedness between xandy in ϑ is given as follows:

Tξ∞(xy)=sup{Tξk(xy):k=1,2,….n}where,Tξk(xy)=min{Tξ(x,v1),TB(v1,v2),…….,Tξ(vn−1,y)}.

Iξ∞(xy)=Inf{Iξk(xy):k=1,2,….n}where,Iξk(xy)=max{Iξ(x,v1),IB(v1,v2),…….,Iξ(vn−1,y)}

²³: A NG, Sn=(V,E) is said to be a neutrosophic star graph if: V={v0,v1,v2,………,vn} with v0 being the central vertex and E={e=(v0vi/1≤i≤n} the edges incident to the central vertex v₀ that connects it to the remaining vertices vi for i=1,2,……..,n .

^27–29: Let G~=(ϑ,ξ ) a connected SVNG, G~ is said to be:

A Turiyam graph TG~=(ϑ,ξ ) of graph G=(V,E) is graph whose vertices and edges must satisfy the following:

ξt(uv)≤ϑt(u)∧ϑt(v),ξi(uv)≤ϑi(u)∧ϑi(v),ξf(uv)≤ϑf(u)˅ϑf(v) and ξl((uv)≤ϑl(u)∧ϑl(v),∀u,v∈V such that 0≤ξt(uv)+ξi(uv)+ξf(uv)+ξl(uv)≤4,∀(uv)∈E.

In TG~ =(ϑ,ξ ), ϑ represents the Turiyam vertex set and ξ is the Turiyam edge set of TG~

Let TG~ =(ϑ,ξ ), be a Turiyam graph. Then, a path P(xxn) in TG~ is a sequence of distinct vertices x,x1,x2,x3……..xn such that ξt(xn−1,xn) > 0, ξi(xn−1,xn) > 0, ξf(xn−1,xn) > 0, and ξl(xn−1,xn) > 0, for 0 ≤ 𝑖 ≤ 𝑛. In this case, 𝑛 ≥ 0 is called the length of path 𝑝. The successive pairs (xn−1,xn) , 0 ≤ 𝑖 ≤ 𝑛, are edges of the path p.

In the following section, we present the certain families of Turiyam graph with some illustrations.

• • Lξ∞(xy)=Inf{Lξk(xy):k=1,2,….n} , and Lξk(x,y)=max{Lξ(x,v1),Lξ(v1,v2),…….Lξ(vn−1,y)} .Where x,v1,.,..vn−1,y∈V

• i. A Turiyam -tree if it has a turiyam spanning sub graph H = (A, C) which is a tree, where for all edges xy not in H satisfying Tξ(x,y)<TC∞(x,y) , Iξ(x,y)>IC∞(x,y) Fξ(x,y)>FC∞(x,y) , and Lξ(x,y)>LC∞(x,y) .

• ii. A Turiyam - forest if 𝐺 has Turiyam spanning sub graph 𝐻 = (𝐴, 𝐶) which is a forest such that, for all ( x,y ) ∉H , satisfying Tξ(x,y)<TC∞(x,y) , Iξ(x,y)>IC∞(x,y) ,

  Fξ(x,y)>FC∞(x,y) , and Lξ(x,y)>LC∞(x,y) .

● Vertex — Edge vi : vertex label { x1,x2,x3,x4 }: 4-tupple values associated with the vertex/edge.

● Vertex — Edge ui : vertex label.

● Vertex — Edge ui : vertex label.

● Vertex — Edge vi : vertex label.

● Vertex — Edge ui : vertex label { x1,x2,x3,x4 }: 4-tupple values associated with the vertex/edge.

● Vertex — Edge ui : vertex label { x1,x2,x3,x4 }: 4-tupple values associated with the vertex/edge.

● Vertex — Edge ui : vertex label { x1,x2,x3,x4 }: 4-tupple values associated with the vertex/edge.