The Mask of Odd Points n

We present an explicit formula for the mask of odd points n-ary, for any odd n⩾3, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-point a-ary schemes introduced by Lian, 2008, and (2m+1)-point a-ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd point n-ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented

Molecular Properties of Carbon Crystal Cubic Structures

Graph theory assumes an imperative part in displaying and planning any synthetic structure or substance organizer. Chemical graph theory facilitates in conception of the chemical graphs for their atomic properties. The graphical structure of a chemical involves atoms termed as vertices and the line segment between two different vertices are called edges. In this manuscript, our concentration is on the chemical graph of carbon graphite and cubic carbon. Additionally, we also define a procedure and calculate the degree based topological indices namely Zagreb type indices, Balaban, Forgotten and Augmented indices

Topological Characterization of Carbon Graphite and Crystal Cubic Carbon Structures

Graph theory is used for modeling, designing, analysis and understanding chemical structures or chemical networks and their properties. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this article, we study the chemical graphs of carbon graphite and crystal structure of cubic carbon. Moreover, we compute and give closed formulas of degree based additive topological indices, namely hyper-Zagreb index, first multiple and second multiple Zagreb indices, and first and second Zagreb polynomials

Laplacian Spectra for Categorical Product Networks and Its Applications

The Kirchhoff index, global mean-first passage time, average path length and number of spanning trees are of great importance in the field of networking. The “Kirchhoff index” is known as a structure descriptor index. The “global mean-first passage time” is known as a measure for nodes that are quickly reachable from the whole network. The “average path length” is a measure of the efficiency of information or mass transport on a network, and the “number of spanning trees” is used to minimize the cost of power networks, wiring connections, etc. In this paper, we have selected a complex network based on a categorical product and have used the spectrum approach to find the Kirchhoff index, global mean-first passage time, average path length and number of spanning trees. We find the expressions for the product and sum of reciprocals of all nonzero eigenvalues of a categorical product network with the help of the eigenvalues of the path and cycles

Blockchain Technology Performance of Asymmetric Algorithms: An Empirical Study

Blockchain technology is a transparent, and unchangeable distributed ledger. It has the potential to transform the way to interact with the digital world by allowing to construct a decentralized database that is tamper-proof. Concerns regarding the security of confidential and sensitive data being outsourced are rising. It is possible that service providers may be dishonest since unscrupulous administrators have access to, may alter, and can misuse private and sensitive data. Security precautions are necessary because sensitive data stored on public clouds has to be safeguarded. There is no mechanism to detect data changes, as data are stored in plaintext. Therefore, maintaining privacy and secrecy is impossible. It is important that data must always be kept secure, even after it has been kept on the server. Data stored on the server must be safeguarded against outsider access and, if the insider cannot be trusted, must also be safeguarded against hostile insiders. Asymmetric algorithms are employed to safeguard data during transmission. Asymmetric cryptography is required in modern security systems, and several algorithms have been devised to provide safe and effective encryption and decryption. Asymmetric algorithms are empirically compared in this study. We evaluated each algorithm's performance by taking into account criteria such as key size, memory utilization, and execution time. Our results show that while all algorithms provide safe encryption and decryption, there are significant performance disparities between them. It is determined, in particular, that ECC required the least amount of memory and had the shortest key size. The findings show that ECC's prime and binary fields created pairs of keys faster and with more security than other asymmetric algorithms with smaller bit sizes. On small, medium, and big datasets, ECC had the fastest execution time for plaintext encryption operations. These findings have important implications for the selection and deployment of asymmetric algorithms in various security systems