sets of vertices for fuzzy(neutrosophic) graph G

prp40

cor41

prp42 1.1. Definitions

• Consider Figure (1.2). Neutrosophic order of N1, On(N1) is (2.57, 2.05, 1.04). Thus On(N1) = (2.57, 2.05, 1.04).

• Consider Figure (1.3). Neutrosophic order of N1, On(N1) is (2.57, 2.05, 1.04). Thus On(N1) = (2.57, 2.05, 1.04).

• Consider Figure (1.4). Neutrosophic order of N1, On(N1) is (2.47, 2.26, 1.47). Thus On(N1) = (2.47, 2.26, 1.47).

• Consider Figure (1.5). Neutrosophic order of N1, On(N1) is (2.22, 1.92, 1.47). Thus On(N1) = (2.47, 2.26, 1.38).

• Consider Figure (1.6). Neutrosophic order of N1, On(N1) is (1.48, 1.28, 0.92). Thus On(N1) = (1.48, 1.28, 0.92).

• Consider Figure (1.7). Neutrosophic order of N1, On(N1) is (1.73, 1.49, 1.13). Thus On(N1) = (1.73, 1.49, 1.13).

Proposition 1.1.40. |N |n ≤ (|N |c, |N |c, |N |c).

Proof.

|N |n = Σn∈N σ(n) = Σn=(n1,n2,n3)∈N (σ(n1), σ(n2), σ(n3)) ≤ Σn=(n1,n2,n3)∈N (1, 1, 1) = (|N |c, |N |c, |N |c).

Corollary 1.1.41. On(N ) ≤ (Oc(N ), Oc(N ), Oc(N )).

Proof. By Proposition (1.1.40), Oc(N ) = |V |c and On(N ) = |V |n, the result is straightforward. Since

On(N ) = |V |n = Σv∈V σ(v) = Σv=(v1,v2,v3)∈V (σ(v1), σ(v2), σ(v3)) ≤ Σn=(v1,v2,v3)∈V (1, 1, 1) = (|V |c, |V |c, |V |c) = (Oc(N ), Oc(N ), Oc(N )).

Proposition 1.1.42. |N |n = (|N |f , |N |f , |N |f ).

Proof.

|N |n = Σn∈N σ(n) = Σn=(n1,n2,n3)∈N (σ(n1), σ(n2), σ(n3)) = (|N |f , |N |f , |N |f ).

In Example (1.1.39), the computations of this notion when they come to neutrosophic order, are done. There’s same type-result with analogous to Corollary (1.1.41). Corollary 1.1.43. On(N ) = (Of (N ), Of (N ), Of (N )).

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