# M-Polar Fuzzy Graphs and Deep Learning for the Design of Analog Amplifiers

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## Abstract

**:**

## 1. Introduction

## 2. Fuzzy Graphs Theory

#### 2.1. Basic Concepts of Fuzzy Graphs

**Definition**

**1**

**.**A fuzzy graph $G=\left(V,\sigma ,\mu \right)$ is a triple consisting of a nonempty set $V$ together with a pair of functions $\sigma :V\to \left[0,1\right]$ and $\mu :E\to \left[0.1\right]$ such that for all $x,y\in V$, $\mu \left(xy\right)\le \sigma \left(x\right)\wedge \sigma \left(y\right)$, where $\wedge $ stands for the minimum.

**Definition**

**2**

**.**A fuzzy graph $G=\left(V,\sigma ,\mu \right)$ is complete if $\mu \left(xy\right)=\sigma \left(x\right)\wedge \sigma \left(y\right)$ for all $x,y\in V$.

**Definition**

**3**

**.**An m-polar fuzzy set (or a ${\left[0,1\right]}^{m}$ -set) on $X$ is mapping $M:X\to {\left[0,1\right]}^{m}$.

**Definition**

**4**

**.**Let $\sigma $ be an m-polar fuzzy set on a set $V$. An m-polar fuzzy relation on $\sigma $ is an m-polar fuzzy set $\mu $ of $V\times V$ such that $\mu \left(xy\right)\le \sigma \left(x\right)\wedge \sigma \left(y\right)$ for all $x,y\in V$, i.e., for each $i=1,2,\dots ,m$, for all $x,y\in V$: ${P}_{i}\circ \mu \left(xy\right)\le {P}_{i}\circ \sigma \left(x\right)\wedge {P}_{i}\circ \sigma \left(y\right)$.

**Definition**

**5**

**.**An m-polar fuzzy graph $G=\left(V,\sigma ,\mu \right)$ is a triple consisting of a nonempty set $V$ together with a pair of functions $\sigma :V\to {\left[0,1\right]}^{m}$ and $\mu :E=V\times V$ →${\left[0,1\right]}^{m}$, where $\sigma $ is an m-polar fuzzy set on the set of vertices $V$ and $\mu $ is an m-polar fuzzy relation in $V$ such that for all $x,y\in V$, $\mu \left(xy\right)\le \sigma \left(x\right)\wedge \sigma \left(y\right)$, where $\wedge $ stands for minimum.

#### 2.2. Products in m-Polar Fuzzy Graphs

#### 2.2.1. Direct (Tensor) Product

**Definition**

**6**

**.**Let ${G}_{1}=\left({\sigma}_{1},{\mu}_{1}\right)$ of ${G}_{1}^{*}=\left({V}_{1},{E}_{1}\right)$ and ${G}_{2}=\left({\sigma}_{2},{\mu}_{2}\right)$ of ${G}_{2}^{*}=\left({V}_{2},{E}_{2}\right)$ be two m-polar fuzzy graphs. The direct product of ${G}_{1}$ and ${G}_{2}$ is denoted by ${G}_{1}\times {G}_{2}$ and is defined as a pair ${G}_{1}\times {G}_{2}=\left({\sigma}_{1}\times {\sigma}_{2},{\mu}_{1}\times {\mu}_{2}\right)$, such that for each $i=1,2,\dots ,m$:

#### 2.2.2. Semi-Strong Product

**Definition**

**7**

**.**Let ${G}_{1}=\left({\sigma}_{1},{\mu}_{1}\right)$ of ${G}_{1}^{*}=\left({V}_{1},{E}_{1}\right)$ and ${G}_{2}=\left({\sigma}_{2},{\mu}_{2}\right)$ of ${G}_{2}^{*}=\left({V}_{2},{E}_{2}\right)$ be two m-polar fuzzy graphs. The semi-strong product of ${G}_{1}$ and ${G}_{2}$ (denoted by ${G}_{1}\u22a1{G}_{2}$) is defined as a graph ${G}_{1}\u22a1{G}_{2}=\left({\sigma}_{1}\u22a1{\sigma}_{2},{\mu}_{1}\u22a1{\mu}_{2}\right)$ of ${G}^{*}=\left({V}_{1}\times {V}_{2},E\right)$ (here $E=\left\{\left({x}_{1},{y}_{1}\right)\left({x}_{1},{y}_{2}\right)\mid {x}_{1}\in {V}_{1},{y}_{1}{y}_{2}\in {E}_{2}\right\}\cup \left\{\left({x}_{1},{y}_{1}\right)\left({x}_{2},{y}_{2}\right)\mid {x}_{1}{x}_{2}\in {E}_{1},{y}_{1}{y}_{2}\in {E}_{2}\right\}$) such that for each $i=1,2,\dots ,m$:

#### 2.2.3. Strong Product

**Definition**

**8**

**.**Let ${G}_{1}=\left({\sigma}_{1},{\mu}_{1}\right)$ of ${G}_{1}^{*}=\left({V}_{1},{E}_{1}\right)$ and ${G}_{2}=\left({\sigma}_{2},{\mu}_{2}\right)$ of ${G}_{2}^{*}=\left({V}_{2},{E}_{2}\right)$ be two m-polar fuzzy graphs. The strong product of ${G}_{1}$ and ${G}_{2}$ (denoted by ${G}_{1}\otimes {G}_{2})$ is defined as a graph ${G}_{1}\otimes {G}_{2}=\left({\sigma}_{1}\otimes {\sigma}_{2},{\mu}_{1}\otimes {\mu}_{2}\right)$ of ${G}^{*}=\left({V}_{1}\times {V}_{2},E\right)$ where

#### 2.2.4. Lexicographic Product

**Definition**

**9**

**.**Let ${G}_{1}=\left({\sigma}_{1},{\mu}_{1}\right)$ of ${G}_{1}^{*}=\left({V}_{1},{E}_{1}\right)$ and ${G}_{2}=\left({\sigma}_{2},{\mu}_{2}\right)$ of ${G}_{2}^{*}=\left({V}_{2},{E}_{2}\right)$ be two m-polar fuzzy graphs. The lexicographic product of ${G}_{1}$ and ${G}_{2}$ (denoted by ${G}_{1}\u2022{G}_{2}$) is defined as a pair ${G}_{1}\u2022{G}_{2}=\left({\sigma}_{1}\u2022{\sigma}_{2},{\mu}_{1}\u2022{\mu}_{2}\right)$ such that for each $i=1,2,\dots ,m$:

**Proposition**

**4**

## 3. Deep Learning and Applications in Electronic Circuit Design

## 4. Proposed Method

- In the first stage, a dataset is prepared according to a predefined specification regarding the designed amplifier. All possible variants of the designed electronic circuit are found and membership functions of attributes are predicted through a deep learning algorithm.
- The second stage points out the suitable design solutions, considering the requested parameters, and after obtaining the membership values of vertices and edges, an m-polar fuzzy graph is constructed.
- In the third stage, the most suitable solutions are prioritized, finding the best one, according to the user’s specifications and certain requirements.

## 5. Experimentation and Results

#### 5.1. Design of Inverting Summing Amplifier

#### 5.2. Design of Subtracting Amplifier (Differential Amplifier)

#### 5.3. Summing and Subtracting Amplifier

## 6. Conclusions

- The synergetic combination of m-polar fuzzy graphs theory and DL leads to obtaining the most suitable solutions only in three stages, extremely reducing the number of repetitive tasks concerning the calculation of the values of designs’ attributes, their comparison, and design selection.
- DL is a suitable approach when expert opinion could be predicted and used for further analysis. In this work, the membership functions of attributes are predicted instead of the expert votes to be gathered. The created predictive models are evaluated, and it is proved that they are characterized with high precision since the obtained errors are very small: RMSE is from 0.0032 to 0.0187, AE is from 0.022 to 0.098, and RE is between 0.27% and 1.57%.
- Fuzzy graph construction gives a possibility for very fast finding the eligible designs, proposes apparatus for their prioritization, and an opportunity for reaching the best design according to a given predefined user specification.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Inverting summing amplifier [44].

**Figure 3.**Prediction chart of the power dissipation ${P}_{D}$ membership values (sigma) at inverting summing amplifier design.

**Figure 7.**Subtracting amplifier [44].

**Figure 8.**The prediction chart of the membership values of power dissipation (sigma) at subtracting amplifier design.

**Figure 12.**Summing and subtracting amplifier [44].

**Figure 13.**The prediction chart of the membership values of power dissipation (sigma) at design of summing and subtracting amplifier.

S | ${\mathit{R}}_{\mathit{F}},\mathit{k}\mathit{\Omega}$ | ${\mathit{R}}_{1},\mathit{k}\mathit{\Omega}$ | ${\mathit{R}}_{2},\mathit{k}\mathit{\Omega}$ | ${\mathit{v}}_{1},\mathit{V}$ | ${\mathit{v}}_{2},\mathit{V}$ | ${\mathit{v}}_{\mathit{o}\mathit{u}\mathit{t}},\mathit{V}$ | ${\mathit{R}}_{\mathit{L}},\mathit{k}\mathit{\Omega}$ | ${\mathit{P}}_{\mathit{D}},\mathit{m}\mathit{W}$ | $\mathit{\sigma}\left({\mathit{P}}_{\mathit{D}}\right)$ |
---|---|---|---|---|---|---|---|---|---|

${S}_{1}$ | 60 | 20 | 10 | 0.01 | 0.01 | 0.09 | 10 | 10.511 | 0.999 |

${S}_{2}$ | 60 | 20 | 10 | 0.05 | 0.01 | 0.21 | 10 | 10.590 | 0.992 |

${S}_{3}$ | 60 | 20 | 10 | 0.1 | 0.01 | 0.36 | 10 | 10.684 | 0.983 |

${S}_{4}$ | 60 | 20 | 10 | 0.15 | 0.01 | 0.51 | 10 | 10.772 | 0.975 |

${S}_{5}$ | 60 | 20 | 10 | 0.2 | 0.01 | 0.66 | 10 | 10.855 | 0.968 |

… | … | … | … | … | … | … | … | … |

$\mathit{S}{\mathit{S}}_{\mathit{i}}\otimes \mathit{S}{\mathit{S}}_{\mathit{k}}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | ${\mathit{A}}_{6}$ | ${\mathit{A}}_{7}$ | ${\mathit{A}}_{8}$ |
---|---|---|---|---|---|---|---|---|

${E}_{12}=S{S}_{1}\otimes S{S}_{2}$ | 0.1 | 0.1 | 0.1 | 1 | 1 | 1 | 0.4 | 0.999 |

${E}_{23}=S{S}_{2}\otimes S{S}_{3}$ | 0.12 | 0.12 | 0.12 | 1 | 1 | 1 | 0.4 | 0.999 |

${E}_{34}=S{S}_{3}\otimes S{S}_{4}$ | 0.1 | 0.1 | 0.1 | 1 | 1 | 1 | 0.4 | 0.999 |

${E}_{45}=S{S}_{4}\otimes S{S}_{5}$ | 0.1 | 0.1 | 0.1 | 1 | 1 | 1 | 0.444 | 0.999 |

${E}_{15}=S{S}_{1}\otimes S{S}_{5}$ | 0.1 | 0.1 | 0.1 | 1 | 1 | 1 | 0.4 | 0.999 |

… | … | … | … | … | … | … | … | … |

$\mathit{S}{\mathit{S}}_{\mathit{i}}\otimes \mathit{S}{\mathit{S}}_{\mathit{k}}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | ${\mathit{A}}_{6}$ | ${\mathit{A}}_{7}$ | ${\mathit{A}}_{8}$ |
---|---|---|---|---|---|---|---|---|

${E}_{12}=S{S}_{1}\otimes S{S}_{2}$ | 1 | 1 | 0.222 | 0.3 | 1 | 1 | 1 | 1 |

${E}_{23}=S{S}_{2}\otimes S{S}_{3}$ | 1 | 1 | 0.25 | 0.333 | 1 | 1 | 1 | 1 |

${E}_{34}=S{S}_{3}\otimes S{S}_{4}$ | 1 | 1 | 0.285 | 0.375 | 1 | 1 | 1 | 1 |

${E}_{45}=S{S}_{4}\otimes S{S}_{5}$ | 1 | 1 | 0.333 | 0.428 | 1 | 1 | 1 | 1 |

${E}_{56}=S{S}_{5}\otimes S{S}_{6}$ | 1 | 1 | 0.4 | 0.5 | 1 | 1 | 1 | 1 |

${E}_{67}=S{S}_{6}\otimes S{S}_{7}$ | 1 | 1 | 0.5 | 0.6 | 1 | 1 | 1 | 1 |

${E}_{78}=S{S}_{7}\otimes S{S}_{8}$ | 1 | 1 | 0.666 | 0.75 | 1 | 1 | 1 | 1 |

… | … | … | … | … | … |

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**MDPI and ACS Style**

Ivanova, M.; Durcheva, M.
M-Polar Fuzzy Graphs and Deep Learning for the Design of Analog Amplifiers. *Mathematics* **2023**, *11*, 1001.
https://doi.org/10.3390/math11041001

**AMA Style**

Ivanova M, Durcheva M.
M-Polar Fuzzy Graphs and Deep Learning for the Design of Analog Amplifiers. *Mathematics*. 2023; 11(4):1001.
https://doi.org/10.3390/math11041001

**Chicago/Turabian Style**

Ivanova, Malinka, and Mariana Durcheva.
2023. "M-Polar Fuzzy Graphs and Deep Learning for the Design of Analog Amplifiers" *Mathematics* 11, no. 4: 1001.
https://doi.org/10.3390/math11041001