A novel topological index, the face index ( F I ), is proposed in this paper. For a molecular graph G, face index is defined as F I ( G ) = ∑ f ∈ F ( G ) d ( f ) = ∑ v ∼ f , f ∈ F ( G ) d ( v ) , where d ( v ) is the degree of the vertex v. The index is very easy to calculate and improved the previously discussed correlation models for π - e l e c t r o n energy and boiling point of benzenoid hydrocarbons. The study shows that the multiple linear regression involving the novel topological index can predict the π -electron energy and boiling points of the benzenoid hydrocarbons with correlation coefficient r > 0.99 . Moreover, the face indices of some planar molecular structures such as 2-dimensional graphene, triangular benzenoid, circumcoronene series of benzenoid are also investigated. The results suggest that the proposed index with good correlation ability and structural selectivity promised to be a useful parameter in QSPR/QSAR.

Topological index is a number that can be used to characterize the graph of a molecule. Topological indices describe the physical, chemical, and biological properties of a chemical structure. In this paper, we derive the analytical closed formulas of face index of some planar molecular structures such as TUC4, TUC4C8S, TUHC6, TUC4C8R, and armchair TUVC6.

With the advent of communication networks, protecting data from security threats has become increasingly important. To address this issue, we present a new text encryption scheme that uses a combination of elliptic curve cryptography and max-plus algebra-based wavelet transform to provide enhanced security and efficiency. The proposed encryption process consists of three main phases. In the first phase, the plaintext is encoded using ASCII characters, followed by the introduction of diffusion in its representation. In the second phase, points are computed on an elliptic curve, and a mapping method is applied to introduce randomness into the data. Finally, in the third phase, the output is decomposed using a max-plus algebra-based wavelet transform to generate the ciphertext. We conduct a comprehensive security analysis of our scheme that includes NIST analysis, entropy analysis, correlation analysis, key space, key sensitivity, plaintext sensitivity, encryption quality, ciphertext-only attack, known-plaintext attack, chosen-plaintext attack, and chosen-ciphertext attack. The findings indicate that the proposed scheme exhibits excellent encryption quality, surpassing a value of 76, which is closer to the ideal value. Moreover, the sensitivity of the plaintext is greater than 91%, indicating its high sensitivity. The correlation between the plaintext and ciphertext is very close to the ideal value of zero. The encrypted texts exhibit a high level of randomness and meet the necessary criteria for a strong key space. These characteristics contribute to its superior security, providing protection against various cryptographic attacks. Additionally, the encryption process for a 5995-character plaintext only takes 0.047 s, while decryption requires 0.038 s. Our results indicate that the proposed scheme offers high levels of security while maintaining reasonable computational efficiency. Thus, it is suitable for secure text communication in various applications. Moreover, when compared with other state-of-the-art text encryption methods, our proposed scheme exhibits better resistance to modern cryptanalysis.