# Laplacian Spectra for Categorical Product Networks and Its Applications

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**Definition**

**1.**

**Lemma**

**1.**

## 3. Main Results

#### 3.1. Categorical Product of Two Paths and Laplacian Spectra

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

#### 3.2. Categorical Product of the Cycle-Path and Laplacian Spectra

**Theorem**

**2.**

**Proof.**

#### 3.3. Categorical Product of the Cycle-Cycle and Laplacian Spectra

**Theorem**

**3.**

**Proof.**

## 4. Applications of Laplacian Spectra

#### 4.1. Kirchhoff Index

#### 4.2. Global Mean-First Passage Time

#### 4.3. Average Path Length

#### 4.4. The Number of Spanning Trees

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Path ${P}_{n}$. (

**b**) Grid ${P}_{n}\times {P}_{2}$ after the first iteration for $t=2$. (

**c**) General grid ${P}_{n}\times {P}_{t}$ structure.

**Figure 3.**(

**a**) Cycle ${C}_{n}$. (

**b**) Grid ${C}_{n}\times {P}_{2}$ after the first iteration for $t=2$. (

**c**) General grid ${C}_{n}\times {P}_{t}$ structure.

**Figure 4.**(

**a**) Initial product ${C}_{n}\times {C}_{3}$ for $t=1$. (

**b**) Grid ${C}_{n}\times {C}_{4}$ after the first iteration for $t=2$.

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**MDPI and ACS Style**

Kang, S.M.; Siddiqui, M.K.; Rehman, N.A.; Imran, M.; Muhammad, M.H.
Laplacian Spectra for Categorical Product Networks and Its Applications. *Symmetry* **2018**, *10*, 206.
https://doi.org/10.3390/sym10060206

**AMA Style**

Kang SM, Siddiqui MK, Rehman NA, Imran M, Muhammad MH.
Laplacian Spectra for Categorical Product Networks and Its Applications. *Symmetry*. 2018; 10(6):206.
https://doi.org/10.3390/sym10060206

**Chicago/Turabian Style**

Kang, Shin Min, Muhammad Kamran Siddiqui, Najma Abdul Rehman, Muhammad Imran, and Mehwish Hussain Muhammad.
2018. "Laplacian Spectra for Categorical Product Networks and Its Applications" *Symmetry* 10, no. 6: 206.
https://doi.org/10.3390/sym10060206