# Certain Properties of Vague Graphs with a Novel Application

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

- (i)
- The degree of a node m is defined as ${d}_{G}\left(m\right)=\left(\right)open="("\; close=")">{d}_{t}\left(m\right),{d}_{f}\left(m\right)$, where ${d}_{t}\left(m\right)={\sum}_{m\ne n}{t}_{B}\left(mn\right)$ and ${d}_{f}\left(m\right)={\sum}_{m\ne n}{f}_{B}\left(mn\right)$.
- (ii)
- The total degree of a node m is defined by $t{d}_{G}\left(m\right)=\left(\right)open="("\; close=")">t{d}_{t}\left(m\right),t{d}_{f}\left(m\right)$, where $t{d}_{t}\left(m\right)={\sum}_{m\ne n}{t}_{B}\left(mn\right)+{t}_{A}\left(m\right)$ and $t{d}_{f}\left(m\right)={\sum}_{m\ne n}{f}_{B}\left(mn\right)+{f}_{A}\left(m\right)$.

**Definition**

**4**

- (i)
- $\overline{V}=V$
- (ii)
- ${\overline{t}}_{A}\left(m\right)={t}_{A}\left(m\right)$, ${\overline{f}}_{A}\left(m\right)={f}_{A}\left(m\right)$, $\forall m\in V$,
- (iii)
- ${\overline{t}}_{B}\left(mn\right)=\left(\right)open="\{"\; close>\begin{array}{cc}0\hfill & if\phantom{\rule{3.33333pt}{0ex}}{t}_{B}\left(mn\right)0,\hfill \\ min({t}_{A}\left(m\right),{t}_{A}\left(n\right))\hfill & if\phantom{\rule{3.33333pt}{0ex}}{t}_{B}\left(mn\right)=0,\hfill \end{array}$

**Definition**

**5**

- (i)
- G is irregular if there is a node neighboring the nodes with distinct degrees.
- (ii)
- G is (TI), if there is a node neighboring the nodes with distinct total degrees.

## 3. New Concepts of IVGs

**Definition**

**6**

- (i)
- G is said to be a highly irregular vague graph (HIVG) if every node of G is a neighbor to vertices with distinct neighborhood degrees.
- (ii)
- G is assumed to be a neighborly irregular vague graph (NIVG) if every two neighbor nodes of G have distinct degrees.
- (iii)
- G is said to be a strongly irregular vague graph (SIVG) if every pair of nodes in G has distinct degrees.

**Example**

**1.**

**Corollary**

**1.**

**Proof.**

**Example**

**2.**

**Theorem**

**1.**

**Proof.**

**Example**

**3.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Remark**

**1.**

**Theorem**

**5.**

**Proof.**

**Example**

**5.**

**Definition**

**7.**

- (i)
- G is called a HTIVG if each node of G is a neighbor to vertices with different neighborhood total degrees.
- (ii)
- G is said to be a neighborly totally irregular vague graph if each two neighbor nodes of G have distinct total degrees.
- (iii)
- G is said to be a STIVG if each pair of node in G has distinct total degrees.

**Example**

**6.**

**Example**

**7.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Example**

**8.**

**Theorem**

**8.**

**Proof.**

**Definition**

**8.**

**Example**

**9.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

**Definition**

**9.**

- (i)
- A NEIVG if each pair of neighbor edges has distinct degrees.
- (ii)
- A neighborly edge totally irregular vague graph (NETIVG) if each pair of neighbor edges has distinct total degrees.

**Example**

**10.**

**Definition**

**10.**

- (i)
- The degree of an edge $mn$ is defined as ${d}_{G}\left(mn\right)=({d}_{t}\left(mn\right),{d}_{f}\left(mn\right))$ where ${d}_{t}\left(mn\right)={d}_{t}\left(m\right)+{d}_{t}\left(n\right)-2{t}_{B}\left(mn\right)$ and ${d}_{f}\left(mn\right)={d}_{f}\left(m\right)+{d}_{f}\left(n\right)-2{f}_{B}\left(mn\right)$.
- (ii)
- The total degree of an edge $mn$ is defined as $t{d}_{G}\left(mn\right)=(t{d}_{t}\left(mn\right),t{d}_{f}\left(mn\right))$ where $t{d}_{t}\left(mn\right)={d}_{t}\left(m\right)+{d}_{t}\left(n\right)-{t}_{B}\left(mn\right)={d}_{t}\left(mn\right)+{t}_{B}\left(mn\right)$ and $t{d}_{f}\left(mn\right)={d}_{f}\left(m\right)+{d}_{f}\left(n\right)-{f}_{B}\left(mn\right)={d}_{f}\left(mn\right)+{f}_{B}\left(mn\right)$.

**Example**

**11.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Proof.**

## 4. Laplacian Energy of VGs

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

**Definition**

**14.**

**Definition**

**15.**

**Remark**

**2.**

**Definition**

**16.**

**Example**

**12.**

**Theorem**

**14.**

**Proof.**

**Definition**

**17.**

**Example**

**13.**

**Theorem**

**15.**

**Proof.**

**Theorem**

**16.**

**Proof.**

**Theorem**

**17.**

**Proof.**

**Theorem**

**18.**

**Proof.**

**Definition**

**18.**

**Definition**

**19.**

**Definition**

**20.**

**Theorem**

**19.**

**Definition**

**21.**

**Example**

**14.**

## 5. Numerical Examples

## 6. Application VG to Find the Most Dominant Person in a Hospital

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 10.**Both Neighborly Edge Irregular (NEI) and Neighborly Edge Totally Irregular Vague Graph (NETIVG).

Notation | Meaning |
---|---|

$\zeta $ | Fuzzy Graph |

VG | Vague Graph |

SI | Strongly Irregular |

HI | Highly Irregular |

TI | Totally Irregular |

STI | Strongly Total Irregular |

NEI | Neighborly Edge Irregular |

NETI | Neighborly Edge Totally Irregular |

NIVG | Neighborly Irregular Vague Graph |

HIVG | Highly Irregular Vague Graph |

SIVG | Strongly Irregular Vague Graph |

NTIVG | Neighborly Totally Irregular Vague Graph |

LE | Laplacian Energy |

AM | Adjacency Matrix |

DM | Degree Matrix |

LM | Laplacian Matrix |

OM | Out-Degree Matrix |

DS | Dominating Set |

△ -DS | △-Dominating Set |

VDG | Vague digraph G |

LS | Laplacian Spectrum |

IG | Influence Graph |

$Nodes\phantom{\rule{3.33333pt}{0ex}}\left(A\right)$ | ${v}_{1}$ | ${v}_{2}$ | ${v}_{3}$ | ${v}_{4}$ | ${v}_{5}$ | ${v}_{6}$ | |

${t}_{A}$ | 0.1 | 0.2 | 0.3 | 0.5 | 0.5 | 0.3 | |

${f}_{A}$ | 0.3 | 0.3 | 0.4 | 0.6 | 0.5 | 0.4 | |

$Edges\phantom{\rule{3.33333pt}{0ex}}\left(B\right)$ | ${v}_{1}{v}_{2}$ | ${v}_{1}{v}_{3}$ | ${v}_{1}{v}_{6}$ | ${v}_{2}{v}_{3}$ | ${v}_{3}{v}_{4}$ | ${v}_{4}{v}_{5}$ | ${v}_{5}{v}_{6}$ |

${t}_{B}$ | 0.1 | 0.1 | 0.1 | 0.2 | 0.3 | 0.4 | 0.3 |

${f}_{B}$ | 0.4 | 0.5 | 0.6 | 0.5 | 0.7 | 0.7 | 0.5 |

Name | Designation |
---|---|

H | Head of the hospital |

D | Doctor |

S | Supervisor |

A | Administrative Staff |

F | Financial Manager |

L | Laboratory |

PH | Pharmacy Manager |

**Table 4.**AM corresponding to Figure 19.

D | H | S | A | F | L | PH | |
---|---|---|---|---|---|---|---|

D | (0.0,1.0) | (0.7.0.2) | (0.0,1.0) | (0.4,0.3) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) |

H | (0.0,1.0) | (0.0,1.0) | (0.5,0.3) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.1,0.4) |

S | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.2,0.3) |

A | (0.4,0.3) | (0.4,0.2) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.2,0.3) |

F | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.3,0.4) | (0.2,0.4) |

L | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.2,0.4) |

PH | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) | (0.0,1.0) |

H | D | S | A | F | L | PH | |
---|---|---|---|---|---|---|---|

${t}_{A}$ | 0.9 | 0.9 | 0.6 | 0.5 | 0.4 | 0.4 | 0.3 |

${f}_{A}$ | 0.0 | 0.0 | 0.2 | 0.2 | 0.3 | 0.3 | 0.2 |

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Rao, Y.; Kosari, S.; Shao, Z.
Certain Properties of Vague Graphs with a Novel Application. *Mathematics* **2020**, *8*, 1647.
https://doi.org/10.3390/math8101647

**AMA Style**

Rao Y, Kosari S, Shao Z.
Certain Properties of Vague Graphs with a Novel Application. *Mathematics*. 2020; 8(10):1647.
https://doi.org/10.3390/math8101647

**Chicago/Turabian Style**

Rao, Yongsheng, Saeed Kosari, and Zehui Shao.
2020. "Certain Properties of Vague Graphs with a Novel Application" *Mathematics* 8, no. 10: 1647.
https://doi.org/10.3390/math8101647