Neutrosophic graphs deals with more complex, uncertain problems in real-life applications which provides more flexibility and compatibility than Intuitionistic fuzzy graphs. The aim of this paper is to enrich the efficiency of the network in accordance with productivity and quality. Here we develop two Neutrosophic graphs into a fully connected Neutrosophic network using the product of graphs. Such a type of network is formed from individuals with unique aspects in every field of work among them. This study proposes extending the other graph products and forming a single valued Neutrosophic graph to find the efficient productivity in the flow of information on a single source network of a single valued Neutrosophic network. An Optimal algorithm is proposed and illustrated with an application.

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.

An optimal network refers to a computer or communication network designed, configured, and managed to maximize efficiency, performance, and effectiveness while minimizing cost and resource utilization. In a network design and management context, optimal typically implies achieving the best possible outcomes between various factors. This research investigated the use of fuzzy graph edge coloring for various fuzzy graph operations, and it focused on the efficacy and efficiency of the fuzzy network product using the minimal spanning tree and the chromatic index of the fuzzy network product. As a network made of nodes and vertices, measurement with vertices is a parameter for domination, and edge measurement is a parameter for edge coloring, so we used these two parameters in the algorithm. This paper aims to identify an optimal network that can be established using product outcomes. This study shows a way to find an optimal fuzzy network based on comparative optimal parameter domination and edge coloring, which can be elaborated with applications. An algorithm was generated using an optimal approach, which was subsequently implemented in the form of applications.

The binomial acyclic graph $B_{\overline{0}}$ comprises one node. The binomial acyclic graph B_k-$B_{\overline{k}}$ is an ordered tree-defined repertoire. The binomial acyclic graph $B_{\overline{k}}$ comprises of two binomial acyclic trees $B_{\overline{k-1}}$ that are linked together. The root of one is the leftmost child of the root of the other. In this paper, we introduce the power of three acyclic graphs for every r, non-negative integer, and prove that power of three acyclic graphs is graceful labeling and also show that every binomial and power of three acyclic graphs is cube sum labeling.

A Hosoya polynomial is a polynomial connected to a molecular graph, which is a graph representation of a chemical compound with atoms as vertices and chemical bonds as edges. A graph invariant is the Hosoya polynomial; it is a graph attribute that does not change under graph isomorphism. It provides information about the number of unique non-empty subgraphs in a given graph. A molecular graph's size and branching complexity are determined by a topological metric known as the Wiener index. The Wiener index of each pair of vertices in a molecular network is the sum of those distances. The topological index, one of the various classes of graph invariants, is a real number related to a connected graph's structure .The goal of this article is to compute the Hosoya polynomial of some class of Abid-Waheed graph. Further, this research focused on a C++ algorithm to calculate the wiener index of and . The Wiener index (W∗I) and Hyper-Wiener index (H∗W∗I) are calculated using Hosoya polynomial (H∗-polynomial) of some family of Abid-Waheed graphs and . Illustrations and applications are given to enhance the research work.

The present paper addresses the critical quest for a viable substitute fuel for compression-ignition engines. With rising crude oil prices and the increasing urgency to reduce emissions, our study indicates the substantial potential of nano-additives. Through a succession of extensive experiments, we have discovered their high degree of effectiveness. In our tests, nano-additives were introduced into the fuel, and their effects were closely monitored using sophisticated methods such as gas chromatography. We have identified significant gaps in current research, particularly regarding the timing of nano-additive application and the streamlining of the testing methodology. The findings are quite promising: the addition of novel nano additives not only enhances engine performance but also drastically reduces harmful emissions. The study is particularly noteworthy because it thoroughly examines the use of various nano-additives combined with biodiesel, demonstrating their versatility and suitability across diverse applications. These coatings reduce the amount of heat loss, increase combustion temperatures, and result in significant emission cuts: carbon monoxide emissions were reduced by 20%, hydrocarbons by 24.5%, and nitrogen by a staggering 27.8% over conventional diesel fuel. This study not only moves alternative fuel research in a whole new direction but also sets the stage for more efficient and cleaner diesel engines. With these new solutions, we can make significant strides toward a promising, sustainable energy future.