Star and Wheel on Recognition of Cancer and Neutrosophic SuperHyperGraph with a Specific Type of Independency of SuperHyperVertices

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Star and Wheel on Recognition of Cancer and Neutrosophic SuperHyperGraph with a Specific Type of Independency of SuperHyperVertices

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Henry Garrett^*

This version is not peer-reviewed

Submitted:

11 February 2024

Posted:

13 February 2024

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Abstract

New ideas on the framework of Neutrosophic SuperHyperGraph for different styles of Neutrosophic SuperHyper-Wheel and Neutrosophic SuperHyper-Star are introduced. More instances and more clarifications alongside sufficient references are featured with a specific type of independency of SuperHyperVertices.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

MSC: 5C17; 05C22; 05E45

1. Scientific Research

Referred to Ref. [2].

Theorem 1.

\begin{matrix} | (\sum_{V_{i} \in V^{'}} T (V_{i}), \sum_{V_{i} \in V^{'}} I (V_{i}), \sum_{V_{i} \in V^{'}} F (V_{i})) | \\ = max_{V^{''} \subseteq V_{N S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \forall V_{i} \in V^{''}, \forall V_{j} \in V^{''} : \\ T_{V}^{'} (V_{i}, V_{j} \in E_{i^{'}}) ≰ min {[T_{V} (V_{i}^{'}), T_{V} (V_{j}^{'})]}_{V_{i}^{'}, V_{j}^{'} \in E_{N S H G}}, \\ I_{V}^{'} (V_{i}, V_{j} \in E_{i^{'}}) ≰ min {[I_{V} (V_{i}^{'}), I_{V} (V_{j}^{'})]}_{V_{i}^{'}, V_{j}^{'} \in E_{N S H G}}, \\ and F_{V}^{'} (E_{i^{'}}) (V_{i}, V_{j} \in E_{i^{'}}) ≰ min {[F_{V} (V_{i}^{'}), F_{V} (V_{j}^{'})]}_{V_{i}^{'}, V_{j}^{'} \in E_{N S H G}} \\ } \end{matrix}

Example 1.

Referred to the Figure 1.

Theorem 2.

Neutrosophic SuperHyperWheelStyles-I don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5}}} F (V_{i})) | \\ = (0.76, 0.76, 0.76) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3, 5} \in V^{''}, \forall V_{j = 1, 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ } \end{matrix}

□

Example 2.

Referred to the Figure 2.

Theorem 3.

Neutrosophic SuperHyperWheelStyles-II don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5}}} F (V_{i})) | \\ = (0.73, 0.73, 0.73) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3, 5} \in V^{''}, \forall V_{j = 1, 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ } \end{matrix}

□

Example 3.

Referred to the Figure 3.

Theorem 4.

Neutrosophic SuperHyperWheelStyles-III don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1 - 4}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 4}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 4}}} F (V_{i})) | \\ = (0.92, 0.92, 0.92) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3} \in V^{''}, \forall V_{j = 1, 3} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 4} \in V^{''}, \forall V_{j = 1, 4} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{2} \in V^{''}, \forall V_{3} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[T_{V^{'}} (V_{2}), T_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[I_{V^{'}} (V_{2}), I_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[F_{V^{'}} (V_{2}), F_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ V_{2} \in V^{''}, \forall V_{4} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[T_{V^{'}} (V_{2}), T_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[I_{V^{'}} (V_{2}), I_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[F_{V^{'}} (V_{2}), F_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \end{matrix}

\begin{matrix} V_{4} \in V^{''}, \forall V_{3} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{4}), T_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{4}), I_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{4}), F_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ } \end{matrix}

□

Example 4.

Referred to the Figure (4).

Theorem 5.

Neutrosophic SuperHyperWheelStyles-IV don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1 - 4, 7}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 4, 7}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 4, 7}}} F (V_{i})) | \\ = (1.15, 1.15, 1.15) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3} \in V^{''}, \forall V_{j = 1, 3} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 4} \in V^{''}, \forall V_{j = 1, 4} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 7, 3} \in V^{''}, \forall V_{j = 7, 3} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 7, 4} \in V^{''}, \forall V_{j = 7, 4} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \end{matrix}

\begin{matrix} V_{1} \in V^{''}, \forall V_{7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{1}), T_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{1}), I_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{1}), F_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ V_{2} \in V^{''}, \forall V_{3} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[T_{V^{'}} (V_{2}), T_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[I_{V^{'}} (V_{2}), I_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[F_{V^{'}} (V_{2}), F_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ V_{2} \in V^{''}, \forall V_{4} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[T_{V^{'}} (V_{2}), T_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[I_{V^{'}} (V_{2}), I_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[F_{V^{'}} (V_{2}), F_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ V_{4} \in V^{''}, \forall V_{3} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{4}), T_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{4}), I_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{4}), F_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ } \end{matrix}

□

Example 5.

Referred to the Figure 5.

Theorem 6.

Neutrosophic SuperHyperWheelStyles-V don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1 - 4, 6 - 7}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 4, 6 - 7}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 4, 6 - 7}}} F (V_{i})) | \\ = (1.39, 1.39, 1.39) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3} \in V^{''}, \forall V_{j = 1, 3} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 4} \in V^{''}, \forall V_{j = 1, 4} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \end{matrix}

\begin{matrix} V_{i = 7, 3} \in V^{''}, \forall V_{j = 7, 3} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 7, 4} \in V^{''}, \forall V_{j = 7, 4} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 6, 3} \in V^{''}, \forall V_{j = 6, 3} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{1} \in V^{''}, \forall V_{7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{1}), T_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{1}), I_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{1}), F_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ V_{1} \in V^{''}, \forall V_{6} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.24, 0.24, 0.24) = \\ min {[T_{V^{'}} (V_{1}), T_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.24, 0.24, 0.24) = \\ min {[I_{V^{'}} (V_{1}), I_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{1}), F_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ V_{6} \in V^{''}, \forall V_{7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{1}), T_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{1}), I_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{1}), F_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \end{matrix}

\begin{matrix} V_{2} \in V^{''}, \forall V_{3} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[T_{V^{'}} (V_{2}), T_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[I_{V^{'}} (V_{2}), I_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[F_{V^{'}} (V_{2}), F_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ V_{2} \in V^{''}, \forall V_{4} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[T_{V^{'}} (V_{2}), T_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[I_{V^{'}} (V_{2}), I_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.19, 0.19, 0.19) = \\ min {[F_{V^{'}} (V_{2}), F_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ V_{4} \in V^{''}, \forall V_{3} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{4}), T_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{4}), I_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.24, 0.24, 0.24) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{4}), F_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ } \end{matrix}

□

Example 6.

Referred to the Figure 6.

Theorem 7.

Neutrosophic SuperHyperWheelStyles-VI don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1 - 7}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 7}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 7}}} F (V_{i})) | \\ = (1.65, 1.65, 1.65) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3} \in V^{''}, \forall V_{j = 1, 3} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 4} \in V^{''}, \forall V_{j = 1, 4} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \end{matrix}

\begin{matrix} V_{i = 1, 5} \in V^{''}, \forall V_{j = 1, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 7, 3} \in V^{''}, \forall V_{j = 7, 3} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 7, 5} \in V^{''}, \forall V_{j = 7, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 2, 5} \in V^{''}, \forall V_{j = 2, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 3, 5} \in V^{''}, \forall V_{j = 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 7, 4} \in V^{''}, \forall V_{j = 7, 4} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 6, 3} \in V^{''}, \forall V_{j = 6, 3} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \end{matrix}

\begin{matrix} V_{1} \in V^{''}, \forall V_{7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{1}), T_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{1}), I_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{1}), F_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ V_{1} \in V^{''}, \forall V_{6} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.24, 0.24, 0.24) = \\ min {[T_{V^{'}} (V_{1}), T_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.24, 0.24, 0.24) = \\ min {[I_{V^{'}} (V_{1}), I_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{1}), F_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ V_{6} \in V^{''}, \forall V_{7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{1}), T_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{1}), I_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{1}), F_{V^{'}} (V_{7})]}_{V_{1}, V_{7} \in E_{1}}, \\ V_{2} \in V^{''}, \forall V_{3} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.26, 0.26, 0.26) ≰ (0.19, 0.19, 0.19) = \\ min {[T_{V^{'}} (V_{2}), T_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.26, 0.26, 0.26) ≰ (0.19, 0.19, 0.19) = \\ min {[I_{V^{'}} (V_{2}), I_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.26, 0.26, 0.26) ≰ (0.19, 0.19, 0.19) = \\ min {[F_{V^{'}} (V_{2}), F_{V^{'}} (V_{3})]}_{V_{2}, V_{3} \in E_{2}}, \\ V_{2} \in V^{''}, \forall V_{4} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.26, 0.26, 0.26) ≰ (0.19, 0.19, 0.19) = \\ min {[T_{V^{'}} (V_{2}), T_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.26, 0.26, 0.26) ≰ (0.19, 0.19, 0.19) = \\ min {[I_{V^{'}} (V_{2}), I_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.26, 0.26, 0.26) ≰ (0.19, 0.19, 0.19) = \\ min {[F_{V^{'}} (V_{2}), F_{V^{'}} (V_{4})]}_{V_{2}, V_{4} \in E_{2}}, \end{matrix}

\begin{matrix} V_{4} \in V^{''}, \forall V_{3} \in V^{''} : \\ T_{V}^{'} (E_{2}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{4}), T_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ I_{V}^{'} (E_{2}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{4}), I_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ and F_{V}^{'} (E_{2}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{4}), F_{V^{'}} (V_{3})]}_{V_{4}, V_{3} \in E_{2}}, \\ V_{4} \in V^{''}, \forall V_{5} \in V^{''} : \\ T_{V}^{'} (E_{3}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{4}), T_{V^{'}} (V_{5})]}_{V_{4}, V_{5} \in E_{3}}, \\ I_{V}^{'} (E_{3}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{4}), I_{V^{'}} (V_{5})]}_{V_{4}, V_{5} \in E_{3}}, \\ and F_{V}^{'} (E_{3}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{4}), F_{V^{'}} (V_{5})]}_{V_{4}, V_{5} \in E_{3}}, \\ V_{4} \in V^{''}, \forall V_{6} \in V^{''} : \\ T_{V}^{'} (E_{3}) = (0.26, 0.26, 0.26) ≰ (0.24, 0.24, 0.24) = \\ min {[T_{V^{'}} (V_{4}), T_{V^{'}} (V_{6})]}_{V_{4}, V_{6} \in E_{3}}, \\ I_{V}^{'} (E_{3}) = (0.26, 0.26, 0.26) ≰ (0.24, 0.24, 0.24) = \\ min {[I_{V^{'}} (V_{4}), I_{V^{'}} (V_{6})]}_{V_{4}, V_{6} \in E_{3}}, \\ and F_{V}^{'} (E_{3}) = (0.26, 0.26, 0.26) ≰ (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{4}), F_{V^{'}} (V_{6})]}_{V_{4}, V_{6} \in E_{3}}, \\ V_{6} \in V^{''}, \forall V_{5} \in V^{''} : \\ T_{V}^{'} (E_{3}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{6}), T_{V^{'}} (V_{5})]}_{V_{6}, V_{5} \in E_{3}}, \\ I_{V}^{'} (E_{3}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{6}), I_{V^{'}} (V_{5})]}_{V_{6}, V_{5} \in E_{3}}, \\ and F_{V}^{'} (E_{3}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{6}), F_{V^{'}} (V_{5})]}_{V_{6}, V_{5} \in E_{3}}, \\ } \end{matrix}

□

Example 7.

Referred to the Figure 7.

Theorem 8.

Neutrosophic SuperHyperStarStyles-I don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5}}} F (V_{i})) | \\ = (0.78, 0.78, 0.78) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3, 5} \in V^{''}, \forall V_{j = 1, 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}} \\ } \end{matrix}

□

Example 8.

Referred to the Figure 8.

Theorem 9.

Neutrosophic SuperHyperStarStyles-II don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5 - 6}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5 - 6}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3, 5 - 6}}} F (V_{i})) | \\ = (1.01, 1.01, 1.01) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3, 5} \in V^{''}, \forall V_{j = 1, 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}} \\ V_{i = 3, 5 - 6} \in V^{''}, \forall V_{j = 3, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 6} \in V^{''}, \forall V_{j = 1, 6} \in V^{''} : \end{matrix}

\begin{matrix} T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ } \end{matrix}

□

Example 9.

Referred to the Figure 9.

Theorem 10.

Neutrosophic SuperHyperStarStyles-III don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 6}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 6}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 6}}} F (V_{i})) | \\ = (1.27, 1.27, 1.27) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \end{matrix}

\begin{matrix} V_{i = 1, 3, 5} \in V^{''}, \forall V_{j = 1, 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 4, 5} \in V^{''}, \forall V_{j = 1, 4, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 3, 5 - 6} \in V^{''}, \forall V_{j = 3, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 4, 5 - 6} \in V^{''}, \forall V_{j = 4, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 6} \in V^{''}, \forall V_{j = 1, 6} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ V_{i = 3, 4} \in V^{''}, \forall V_{j = 3, 4} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ } \end{matrix}

□

Example 10.

Referred to the Figure 10.

Theorem 11.

Neutrosophic SuperHyperStarStyles-IV don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 6}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 6}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 6}}} F (V_{i})) | \\ = (1.24, 1.24, 1.24) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3, 5} \in V^{''}, \forall V_{j = 1, 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 4, 5} \in V^{''}, \forall V_{j = 1, 4, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \end{matrix}

\begin{matrix} and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 3, 5 - 6} \in V^{''}, \forall V_{j = 3, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 4, 5 - 6} \in V^{''}, \forall V_{j = 4, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 6} \in V^{''}, \forall V_{j = 1, 6} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ V_{i = 3, 4} \in V^{''}, \forall V_{j = 3, 4} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ } \end{matrix}

□

Example 11.

Referred to the Figure 11.

Theorem 12.

Neutrosophic SuperHyperStarStyles-V don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 6}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 6}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 6}}} F (V_{i})) | \\ = (1.21, 1.21, 1.21) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3, 5} \in V^{''}, \forall V_{j = 1, 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 4, 5} \in V^{''}, \forall V_{j = 1, 4, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 3, 5 - 6} \in V^{''}, \forall V_{j = 3, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \end{matrix}

\begin{matrix} I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 4, 5 - 6} \in V^{''}, \forall V_{j = 4, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 6} \in V^{''}, \forall V_{j = 1, 6} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ V_{i = 3, 4} \in V^{''}, \forall V_{j = 3, 4} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ } \end{matrix}

□

Example 12.

Referred to the Figure 12.

Theorem 13.

Neutrosophic SuperHyperStarStyles-VI don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 7}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 7}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1, 3 - 4, 5 - 7}}} F (V_{i})) | \\ = (1.44, 1.44, 1.44) = \\ = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3, 5} \in V^{''}, \forall V_{j = 1, 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 4, 5} \in V^{''}, \forall V_{j = 1, 4, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 3, 5 - 6} \in V^{''}, \forall V_{j = 3, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \end{matrix}

\begin{matrix} I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 4, 5 - 6} \in V^{''}, \forall V_{j = 4, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 3, 5 - 6} \in V^{''}, \forall V_{j = 3, 5 - 7} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 4, 5 - 6} \in V^{''}, \forall V_{j = 4, 5 - 7} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 6} \in V^{''}, \forall V_{j = 1, 6} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ V_{i = 1, 7} \in V^{''}, \forall V_{j = 1, 7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ V_{i = 6, 7} \in V^{''}, \forall V_{j = 6, 7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \end{matrix}

\begin{matrix} V_{i = 3, 4} \in V^{''}, \forall V_{j = 3, 4} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ } \end{matrix}

□

Example 13.

Referred to the Figure 13.

Theorem 14.

Neutrosophic SuperHyperStarStyles-VII don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i = 1 - 7}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 7}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i = 1 - 7}}} F (V_{i})) | \\ = (1.67, 1.67, 1.67) = \end{matrix}

\begin{matrix} = max_{V^{''} \subseteq V_{S H G}} | {(\sum_{V_{i} \in V^{''}} T (V_{i}), \sum_{V_{i} \in V^{''}} I (V_{i}), \sum_{V_{i} \in V^{''}} F (V_{i})) | \\ V_{i = 1, 3, 5} \in V^{''}, \forall V_{j = 1, 3, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 4, 5} \in V^{''}, \forall V_{j = 1, 4, 5} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 3, 5 - 6} \in V^{''}, \forall V_{j = 3, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 4, 5 - 6} \in V^{''}, \forall V_{j = 4, 5 - 6} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 3, 5 - 6} \in V^{''}, \forall V_{j = 3, 5 - 7} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 4, 5 - 6} \in V^{''}, \forall V_{j = 4, 5 - 7} \in V^{''} : \\ T_{V}^{'} (E_{k} \sim^{NOT DEFINED} min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ I_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ and F_{V}^{'} (E_{k}) \sim^{NOT DEFINED} min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \notin E_{k}}, \\ V_{i = 1, 6} \in V^{''}, \forall V_{j = 1, 6} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ V_{i = 1, 6} \in V^{''}, \forall V_{j = 1, 7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \end{matrix}

\begin{matrix} V_{i = 6, 7} \in V^{''}, \forall V_{j = 6, 7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ V_{i = 3, 4} \in V^{''}, \forall V_{j = 3, 4} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ V_{i = 2} \in V^{''}, \forall V_{j = 1, 3 - 7} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[T_{V^{'}} (V_{i}), T_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[I_{V^{'}} (V_{i}), I_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) ≰ (0.23, 0.23, 0.23) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}} \\ } \end{matrix}

□

Example 14.

Referred to the Figure 14.

References

Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph As Hyper Tool On Super Toot”, Curr Trends Mass Comm 2(1) (2023) 32-55. (https://www.opastpublishers.com/open-access-articles/new-ideas-in-recognition -of-cancer-and-neutrosophic-super-hypergraph-as-hyper-tool-on-super-toot.pdf).
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph As Hyper Tool On Super Toot”, Curr Trends Mass Comm 2(1) (2023) 32-55. (https://www.opastpublishers.com/open-access-articles/new-ideas-in-recognition-of-cancer -and-neutrosophic-super-hypergraph-as-hyper-tool-on-super-toot.pdf).
Henry Garrett, “Some Super Hyper Degrees and Co-Super Hyper Degrees on Neutrosophic Super Hyper Graphs And Super Hyper Graphs Alongside Applications in Cancer’s Treatments”, J Math Techniques Comput Math 2(1) (2023) 35-47. (https://www.opastpublishers.com/open-access-articles/some-super-hyper-degrees-and-cosuper-hyper-degrees -on-neutrosophic-super-hyper-graphs-and-super-hyper-graphs-alongside-a.pdf).
Henry Garrett, “A Research on Cancer’s Recognition and Neutrosophic Super Hypergraph by Eulerian Super Hyper Paths and Hamiltonian Sets as Hyper Covering Versus Super separations”, J Math Techniques Comput Math 2(3) (2023) 136-148. (https://www.opastpublishers.com/open-access-articles/a-research-on-cancers-recognition-and-neutrosophic- super-hypergraph-by-eulerian-super-hyper-Paths-and-hamiltonian-sets-.pdf).
Henry Garrett, “Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction to Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition and Beyond”, J Math Techniques Comput Math 2(6) (2023) 221-307. (https://www.opastpublishers.com/open-access-articles/neutrosophic-1failed-superhyperforcing-in-the- superhyperfunction-to-use-neutrosophic-superhypergraphs-on-cancers-neutros.pdf).
Henry Garrett, “Neutrosophic Co-degree and Neutrosophic Degree alongside Chromatic Numbers in the Setting of Some Classes Related to Neutrosophic Hypergraphs”, J Curr Trends Comp Sci Res 2(1) (2023) 16-24. [CrossRef]
Henry Garrett, “Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes”, J Math Techniques Comput Math 1(3) (2022) 242-263. [CrossRef]
Garrett, Henry. “0039 | Closing Numbers and Super-Closing Numbers as (Dual)Resolving and (Dual)Coloring alongside (Dual)Dominating in (Neutrosophic)n-SuperHyperGraph.” CERN European Organization for Nuclear Research - Zenodo, Nov. 2022. CERN European Organization for Nuclear Research. [CrossRef]
Garrett, Henry. “0049 | (Failed)1-Zero-Forcing Number in Neutrosophic Graphs.” CERN European Organization for Nuclear Research - Zenodo, Feb. 2022. CERN European Organization for Nuclear Research. [CrossRef]
Henry Garrett, “Beyond Neutrosophic Graphs”. Dr. Henry Garrett, 2023. [CrossRef]
Henry Garrett, “Neutrosophic Duality”. Dr. Henry Garrett, 2023. [CrossRef]
Aaron, D. Gray, and Michael A. Henning, “Paired-domination game played on Paths, Discrete Applied Mathematics 336 (2023) 132-146. [CrossRef]
Walter Carballosa, and Justin Wisby, “Total k-domination in Cartesian product of complete graphs, Discrete Applied Mathematics 337 (2023) 25-41. [CrossRef]
R. R. Del-Vecchio, and M. Kouider, New bounds for the b-chromatic number of vertex deleted graphs", Discrete Applied Mathematics 306 (2022) 108-113.
M. E. Elaine et al., Bipartite completion of colored graphs avoiding chordless Paths of given lengths", Discrete Applied Mathematics (2022).
R. Janczewski et al., Infinite chromatic games", Discrete Applied Mathematics 309 (2022) 138-146.
L. Li, and X. Li, Edge-disjoint rainbow triangles in edge- colored graphs", Discrete Applied Mathematics 318 (2022) 21-30.
W. Li et al., Rainbow triangles in arc-colored digraphs", Discrete Applied Mathematics 314 (2022) 169-180.
Z.Lu,andL.Shi,A sufficient condition for edge-colorable planar graphs with maximum degree 6 ", Discrete Applied Mathematics 313 (2022) 67-70.
Z. Masih, and M. Zaker, Some comparative results concerning the Grundy and b-chromatic number of graphs", Discrete Applied Mathematics 306 (2022) 1-6.
F. Wu et al., Color neighborhood union conditions for proper edge-pancyclicity of edge-colored complete graphs", Discrete Applied Mathematics 307 (2022) 145-152.

Figure 1. Referred to the Example (1)

Figure 2. Referred to the Example (2)

Figure 3. Referred to the Example (3)

Figure 4. Referred to the Example (4)

Figure 5. Referred to the Example (5)

Figure 6. Referred to the Example (6)

Figure 7. Referred to the Example (7)

Figure 8. Referred to the Example (8)

Figure 9. Referred to the Example (9)

Figure 10. Referred to the Example (10)

Figure 11. Referred to the Example (14)

Figure 12. Referred to the Example (14)

Figure 13. Referred to the Example (14)

Figure 14. Referred to the Example (14)

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Star and Wheel on Recognition of Cancer and Neutrosophic SuperHyperGraph with a Specific Type of Independency of SuperHyperVertices

Abstract

1. Scientific Research

References

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