Recognition of Cancer With Bipartite and Path as Neutrosophic SuperHyperGraph By the Tool Called Stability

Preprint

Article

Recognition of Cancer With Bipartite and Path as Neutrosophic SuperHyperGraph By the Tool Called Stability

Altmetrics

Downloads

Views

178

Comments

Henry Garrett^*

This version is not peer-reviewed

Submitted:

29 August 2023

Posted:

29 August 2023

You are already at the latest version

Alerts

Abstract

New ideas on the framework of Neutrosophic SuperHyperGraph for different styles of Neutrosophic SuperHyper-Bipartite and Neutrosophic SuperHyper-Path are introduced. More instances and more clarifications alongside sufficient references.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

MSC: 05C17; 05C22; 05E45

1. Scientific Research

Referred to Ref. [2].

Theorem 1.1.

\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^{'}}) \neg \leq 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^{'}}) \neg \leq 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^{'}}) \neg \leq min {[F_{V} (V_{i}^{'}), F_{V} (V_{j}^{'})]}_{V_{i}^{'}, V_{j}^{'} \in E_{N S H G}} \\ } \end{matrix}

Example 1.2.

Referred to the Figure 1.

Theorem 1.3.

Neutrosophic SuperHyperBipartiteStyles-I don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{1}}} T (V_{i}), \sum_{V_{i} \in V^{'} = {V_{1}}} I (V_{i}), \sum_{V_{i} \in V^{'} = {V_{1}}} F (V_{i})) | \\ = (0.26, 0.26, 0.26) . \end{matrix}

□

Example 1.4.

Referred to the Figure 2.

Theorem 1.5.

Neutrosophic SuperHyperBipartiteStyles-II don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{1}, V_{2}}} T (V_{i}), \sum_{V_{i} \in V^{'} = {V_{1}, V_{2}}} I (V_{i}), \sum_{V_{i} \in V^{'} = {V_{1}, V_{2}}} F (V_{i})) | \\ = (0.5, 0.5, 0.5) = \\ = 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_{1} \in V^{''}, \forall V_{2} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ min {[T_{V^{'}} (V_{1}), T_{V^{'}} (V_{2})]}_{V_{1}, V_{2} \in E_{1}}, \\ I_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ min {[I_{V^{'}} (V_{1}), I_{V^{'}} (V_{2})]}_{V_{1}, V_{2} \in E_{1}}, \\ and F_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{1}), F_{V^{'}} (V_{2})]}_{V_{1}, V_{2} \in E_{1}}, \\ } \end{matrix}

□

Example 1.6.

Referred to the Figure 3.

Theorem 1.7.

Neutrosophic SuperHyperBipartiteStyles-III don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{3}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{3}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{3}}} F (V_{i})) | \\ = (0.75, 0.75, 0.75) = \\ = 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} \in V^{''}, \forall V_{j} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ } \end{matrix}

□

Example 1.8.

Referred to the Figure 4.

Theorem 1.9.

Neutrosophic SuperHyperBipartiteStyles-IV don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{4}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{4}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{4}}} F (V_{i})) | \\ = (0.99, 0.99, 0.99) = \\ = 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} \in V^{''}, \forall V_{j} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ } \end{matrix}

□

Example 1.10.

Referred to the Figure 5.

Theorem 1.11.

Neutrosophic SuperHyperBipartiteStyles-V don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{5}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{5}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{5}}} F (V_{i})) | \\ = (1.23, 1.23, 1.23) = \\ = 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} \in V^{''}, \forall V_{j} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ } \end{matrix}

□

Example 1.12.

Referred to the Figure 6.

Theorem 1.13.

Neutrosophic SuperHyperBipartiteStyles-VI don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{6}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{6}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{6}}} F (V_{i})) | \\ = (1.47, 1.47, 1.47) = \\ = 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} \in V^{''}, \forall V_{j} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ } \end{matrix}

□

Example 1.14.

Referred to the Figure 7.

Theorem 1.15.

Neutrosophic SuperHyperBipartiteStyles-VII don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{7}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{7}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{i}}_{i = 1^{7}}} F (V_{i})) | \\ = (1.71, 1.71, 1.71) = \\ = 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} \in V^{''}, \forall V_{j} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ } \end{matrix}

□

Example 1.16.

Referred to the Figure 8.

Theorem 1.17.

Neutrosophic SuperHyperPathStyles-I don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{1}, V_{8}, V_{12}, V_{16}, V_{20}, V_{25}, V_{32}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{1}, V_{8}, V_{12}, V_{16}, V_{20}, V_{25}, V_{32}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{1}, V_{8}, V_{12}, V_{16}, V_{20}, V_{25}, V_{32}}} F (V_{i})) | \\ = (1.82, 1.82, 1.82) \\ = 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, 8, 12, 16, 20, 25, 32} \in V^{''}, \forall V_{j = 1, 8, 12, 16, 20, 25, 32} \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 1.18.

Referred to the Figure 9.

Theorem 1.19.

Neutrosophic SuperHyperPathStyles-II don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{1 - 2, 8, 11 - 12, 15 - 16, 19 - 20, 24 - 25, 32 - 34}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{1 - 2, 8, 11 - 12, 15 - 16, 19 - 20, 24 - 25, 32 - 34}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{1 - 2, 8, 11 - 12, 15 - 16, 19 - 20, 24 - 25, 32 - 34}}} F (V_{i})) | \\ = (3.48, 3.48, 3.48) \\ = 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, 8, 12, 16, 20, 25, 32} \in V^{''}, \forall V_{j = 1, 8, 12, 16, 20, 25, 32} \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 - 2, 11, 15, 19, 24} \in V^{''}, \forall V_{j = 1 - 2, 11, 15, 19, 24} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ V_{i = 32 - 34} \in V^{''}, \forall V_{j = 32 - 34} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ } \end{matrix}

□

Example 1.20.

Referred to the Figure 10.

Theorem 1.21.

Neutrosophic SuperHyperPathStyles-III don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{1 - 2, 8 - 12, 15 - 16, 19 - 20, 24 - 25, 31 - 34}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{1 - 2, 8 - 12, 15 - 16, 19 - 20, 24 - 25, 31 - 34}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{1 - 2, 8 - 12, 15 - 16, 19 - 20, 24 - 25, 31 - 34}}} F (V_{i})) | \\ = (4.2, 4.2, 4.2) \\ = 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, 8, 12, 16, 20, 25, 32} \in V^{''}, \forall V_{j = 1, 8, 12, 16, 20, 25, 32} \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 - 2, 11, 15, 19, 24} \in V^{''}, \forall V_{j = 1 - 2, 11, 15, 19, 24} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ V_{i = 32 - 34} \in V^{''}, \forall V_{j = 32 - 34} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ V_{i = 8 - 10, 31, 2} \in V^{''}, \forall V_{j = 8 - 10, 31, 2} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ } \end{matrix}

□

Example 1.22.

Referred to the Figure 11.

Theorem 1.23.

Neutrosophic SuperHyperPathStyles-IV don’t coincide.

Proof.

\begin{matrix} | (\sum_{V_{i} \in V^{'} = {V_{1 - 4, 8 - 12 - 16, 19 - 20, 24 - 25, 31 - 34}}} T (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{1 - 4, 8 - 12 - 16, 19 - 20, 24 - 25, 31 - 34}}} I (V_{i}), \\ \sum_{V_{i} \in V^{'} = {V_{1 - 4, 8 - 12 - 16, 19 - 20, 24 - 25, 31 - 34}}} F (V_{i})) | \\ = (5.16, 5.16, 5.16) \\ = 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, 8, 12, 16, 20, 25, 32} \in V^{''}, \forall V_{j = 1, 8, 12, 16, 20, 25, 32} \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 - 2, 11, 15, 19, 24} \in V^{''}, \forall V_{j = 1 - 2, 11, 15, 19, 24} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ V_{i = 32 - 34} \in V^{''}, \forall V_{j = 32 - 34} \in V^{''} : \end{matrix}

\begin{matrix} T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ V_{i = 8 - 10, 31, 2} \in V^{''}, \forall V_{j = 8 - 10, 31, 2} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ V_{i = 12 - 14, 3 - 4} \in V^{''}, \forall V_{j = 12 - 14, 3 - 4} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ V_{i = 4, 16} \in V^{''}, \forall V_{j = 4, 16} \in V^{''} : \\ T_{V}^{'} (E_{1}) = (0.26, 0.26, 0.26) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ 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) \neg \leq (0.24, 0.24, 0.24) = \\ min {[F_{V^{'}} (V_{i}), F_{V^{'}} (V_{j})]}_{V_{i}, V_{j} \in E_{1}}, \\ } \end{matrix}

□

Example 1.24.

Referred to the Figure 12.

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. Available online: 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. Available online: 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. Available online: 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. Available online: 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. Available online: 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 2023, 1, 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, 2022; 1, 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. [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. Aaron D. Gray, and Michael A. Henning, Paired-domination game played on Paths. Discrete Applied Mathematics 2023, 336, 132–146. [Google Scholar] [CrossRef]
Walter Carballosa, and Justin Wisby. Total k-domination in Cartesian product of complete graphs. Discrete Applied Mathematics 2023, 337, 25–41. [CrossRef]
R. R. Del-Vecchio, and M. Kouider, New bounds for the b-chromatic number of vertex deleted graphs. Discrete Applied Mathematics 2022, 306, 108–113. [CrossRef]
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 2022, 309, 138–146. [CrossRef]
L. Li, and X. Li, Edge-disjoint rainbow triangles in edge- colored graphs. Discrete Applied Mathematics 2022, 318, 21–30. [CrossRef]
W. Li et al., Rainbow triangles in arc-colored digraphs. Discrete Applied Mathematics 2022, 314, 169–180. [CrossRef]
Z.Lu,andL.Shi,A sufficient condition for edge-colorable planar graphs with maximum degree 6. Discrete Applied Mathematics 2022, 313, 67–70. [CrossRef]
Masih, and M. Zaker, Some comparative results concerning the Grundy and b-chromatic number of graphs. Discrete Applied Mathematics 2022, 306, 1–6. [CrossRef]
F. Wu et al., Color neighborhood union conditions for proper edge-pancyclicity of edge-colored complete graphs. Discrete Applied Mathematics 2022, 307, 145–152. [CrossRef]

Figure 1. Referred to the Example 1.2.

Figure 2. Referred to the Example 1.4.

Figure 3. Referred to the Example 1.6.

Figure 4. Referred to the Example 1.8.

Figure 5. Referred to the Example 1.10.

Figure 6. Referred to the Example 1.12.

Figure 7. Referred to the Example 1.14.

Figure 8. Referred to the Example 1.16.

Figure 9. Referred to the Example (1.18)

Figure 10. Referred to the Example 1.20.

Figure 11. Referred to the Example (1.22)

Figure 12. Referred to the Example 1.24.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

Recognition of Cancer With Bipartite and Path as Neutrosophic SuperHyperGraph By the Tool Called Stability

Abstract

1. Scientific Research

References

MDPI Initiatives

Important Links

Subscribe