The perfect matchings counting problem of graphs has important applications in combinatorial optimization, statistical physics, quantum chemistry and other fields. A perfect matching of a graph G is a set of non-adjacent edges that covers all vertices of G . The number of perfect matchings of a graph is closely related to its number of vertices. A fullerene graph   is a 3-connected cubic planar graphs all of whose faces are pentagons and hexagons. Došlić obtained that a fullerene graph with P  vertices has at least P/2+1  perfect matchings, Zhang et al. proved a better lower bound  3(p+2)/4 of the number of perfect matchings of a fullerene graph. We have known that the fullerene graph has a nontrivial cyclic 5-edge-cut if and only if it is isomorphic to the graph T_n for some integer n >=1, where T_n  is the tubular fullerene graph T_n comprised of two caps formed of six pentagons joined by n concentric layers of hexagons.  In this paper, the perfect matchings of the graph T_n  is classified by matching a certain vertex, and recursive relations of a set of perfect matching numbers are obtained. Then the calculation formula of the number of perfect matchings of the graph T_n  is given by recursive relationships. Finally, we get the number of perfect matchings of  T_n  with P  vertices.

Security of a digital image can be achieved in number of ways including image Encryption and Decryption. Encryption technique tries to convert a plain image into the cipher image which is hard to understand. The decryption technique tries to convert a cipher image into the plain image. The image encryption is done inorder to provide security from attacks. In this work we aim to develop an image encryption technique based on multidimensional chaotic map with genetic operator. One of the powerful features of genetic operator is crossover which is used to confuse the pixels of the image. A combination of multidimensional chaotic maps, namely, Logistic, Henon and Chebyshev will be used to generate pseudorandom sequence which will be XOR'ed to obtain an unpredictable sequence. This sequence will be then applied to the Crossover unit and upon XOR'ed to obtain an encrypted image. Later the same upredictable sequence is generated while decrypting the image. By combining the entire different dimensional chaotic map namely Logistic, Henon, Chebyshev map along with the gentic operator called crossover will enhance the extra security to the digital image.

In this paper, A 6 (six) compartmental (S, IU, IS, IA, Q, R) model was presented to examine the dynamical behavior of disease transmission in the system with quarantine effect on the symptomatic infected, asymptomatic infected and Reproduction number R0 within a given population. The parameters model was analyzed and estimated experimentally using the real data of COVID-19 confirmed cases for Ethiopia via MATLAB 2021a. Reproduction number R0 which is a key indicator to whether a disease outbreak spread force will persist or die out within population. R0 was found using the next generation matrix with Gaussian elimination method to obtain the inverse of the transitive matrix. The model also aims at reducing R0 owning to the fact that when the basic reproduction number is less than 1 infected person, disease dies out and when the reproduction number is greater than 1 infected person, the disease persists. The facts about R0 geared us to mathematically check for the Routh-Hurwitz stability criteria and Lyapunov Functions to concisely establish the necessary and sufficient conditions for the Local and Global stability of model. results show that, when R0 < 1 and R0 > 1 the diseases free equilibrium and endemic equilibrium points are locally and globally asymptotically stable respectively. In order to interpret results and recommend possible control measure of disease, The dynamics of the Quarantine compartment in model was tested via sensitivity analysis to experimentally investigate transition/ transmission pattern. The effect of quarantine analysis on the model shows that preventive measures such as increase in quarantine with treatments during disease outbreak will significantly decrease the Reproduction number. Hence, increase in Quarantine compartment will flatten the curve of (S, IU, IS, IA, Q, R) dynamic model correspondingly.

Since the inception of Blockchain, the computer database has been evolving into innovative technologies. Recent technologies emerge, the use of Blockchain is also flourishing. All the technologies from Blockchain use a mutual algorithm to operate. The consensus algorithm is the process that assures mutual agreements and stores information in the decentralized database of the network. Blockchain’s biggest drawback is the exposure to scalability. However, using the correct consensus for the relevant work can ensure efficiency in data storage, transaction finality, and data integrity. In this paper, a comparison study has been made among the following consensus algorithms: Proof of Work (PoW), Proof of Stake (PoS), Proof of Authority (PoA), and Proof of Vote (PoV). This study aims to provide readers with elementary knowledge about blockchain, more specifically its consensus protocols. It covers their origins, how they operate, and their strengths and weaknesses. We have made a significant study of these consensus protocols and uncovered some of their advantages and disadvantages in relation to characteristics details such as security, energy efficiency, scalability, and IoT (Internet of Things) compatibility. This information will assist future researchers to understand the characteristics of our selected consensus algorithms.

In this short article an attempt has been made to model Monte Carlo simulation to solve integration problems. The Monte Carlo method employs random sampling and the theory of big numbers to generate values that are very close to the integral's true solution. Python programming has been used to implement the developed algorithm for integration. The developed Python functions are tested with the help of six different integration examples which are difficult to solve analytically. It has been observed that that the Monte Carlo simulation has given results which are in good agreement with the exact analytical results.