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Cryptography is a pivotal application of graph theory in ensuring secure communication systems. Modern cryptography is deeply rooted in mathematical theory and computer science practices. It is widely recognized that encryption and decryption processes are primarily outcomes of mathematical research. Given the increasing importance of safeguarding secret information or messages from potential intruders, it is imperative to develop effective technical tools for this purpose. These intruders are often well-versed in the latest technological advancements that could breach security. To address this, our study focuses on the efficacious combinatorial technique of graph networks using efficient domination and total magic labeling. The introduction of a graph network based on total magic labeling can significantly influence the network’s performance. This research introduces a novel combinatorial method for encrypting and decrypting confidential numbers by leveraging an efficient dominant notion and labeled graph.

Subgraph and supergraph search methods are promising techniques for the development of new drugs. For example, the chemical structure of favipiravir—an antiviral treatment for influenza—resembles the structure of some components of RNA. Represented as graphs, such compounds are similar to a subgraph of favipiravir. However, the existing supergraph search methods can only discover compounds that match exactly. We propose a novel problem, called similar supergraph search, and design an efficient algorithm to solve it. The problem is to identify all graphs in a database that are similar to any subgraph of a query graph, where similarity is defined as edit distance. Our algorithm represents the set of candidate subgraphs by a code tree, which it uses to efficiently compute edit distance. With a distance threshold of zero, our algorithm is equivalent to an existing efficient algorithm for exact supergraph search. Our experiments show that the computation time increased exponentially as the distance threshold increased, but increased sublinearly with the number of graphs in the database.

A labeled graph is a special structure with node identification capability, which is often used in information networks, biological networks, and other fields. The subgraph query is widely used as an important means of graph data analysis. As the size of the labeled graph increases and changes dynamically, users tend to focus on the high-match results that are of interest to them, and they want to take advantage of the relationship and number of results to get the results of the query quickly. For this reason, we consider the individual needs of users and propose a dynamic Top-K interesting subgraph query. This method establishes a novel graph topology feature index (GTSF index) including a node topology feature index (NTF index) and an edge feature index (EF index), which can effectively prune and filter the invalid nodes and edges that do not meet the restricted condition. The multi-factor candidate set filtering strategy is proposed based on the GTSF index, which can be further pruned to obtain fewer candidate sets. Then, we propose a dynamic Top-K interesting subgraph query method based on the idea of the sliding window to realize the dynamic modification of the matching results of the subgraph in the dynamic evolution of the label graph, to ensure real-time and accurate results of the query. In addition, considering the factors, such as frequent Input/Output (I/O) and network communication overheads, the optimization mechanism of the graph changes and an incremental maintenance strategy for the index are proposed to reduce the huge cost of redundant operation and global updates. The experimental results show that the proposed method can effectively deal with a dynamic Top-K interesting subgraph query on a large-scale labeled graph, at the same time the optimization mechanism of graph changes and the incremental maintenance strategy of the index can effectively reduce the maintenance overheads.