Competition graph is a graph which constitutes from a directed graph (digraph) with an edge between two vertices if they have some common preys in the digraph. Moreover, Fuzzy competition graph (briefly, FCG) is the higher extension of the crisp competition graph by assigning fuzzy value to each vertex and edge. Also, Interval-valued FCG (briefly, IVFCG) is another higher extension of fuzzy competition graph by taking each fuzzy value as a sub-interval of the interval $[0,1]$ . This graph arises in many real world systems; one of them is discussed as follows: Each and every species in nature basically needs ecological balance to survive. The existing species depends on one another for food. If there happens any extinction of any species, there must be a crisis of food among those species which depend on that extinct species. The height of food crisis among those species varies according to their ecological status, environment and encompassing atmosphere. So, the prey to prey relationship among the species cannot be assessed exactly. Therefore, the assessment of competition of species is vague or shadowy. Motivated from this idea, in this paper IVFCG is introduced and several properties of IVFCG and its two variants interval-valued fuzzy k-competition graphs (briefly, IVFKCG) and interval-valued fuzzy m-step competition graphs (briefly, IVFMCG) are presented. The work is helpful to assess the strength of competition among competitors in the field of competitive network system. Furthermore, homomorphic and isomorphic properties of IVFCG are also discussed. Finally, an appropriate application of IVFCG in the competition among the production companies in market is presented to highlight the relevance of IVFCG.

Many existing studies show that there exists a strong relationship between structures and characteristics of molecules. Topological indices are often used in modeling the properties of chemical compounds and biological activities in theoretical chemistry. Topological indices are numerical values associated with structures of molecules in such a way that they remain constant under graph isomorphism. Multiplicative Zagreb indices are among the famous topological indices that have been explored by numerous researchers in the last few years. The first objective of the present paper is to examine the importance of general multiplicative Zagreb indices for forecasting the enthalpy of formation of hydrocarbons using a data set of 25 benzenoid hydrocarbons. The second objective of this paper is to study molecular trees with a given order and with a given number of branching vertices or segments using general multiplicative (first and second) Zagreb indices. Sharp lower/upper bounds on these Zagreb indices for the aforementioned molecular trees are obtained and the graphs attaining these bounds are determined. Bounds on the classical multiplicative Zagreb and Narumi-Katayama indices are corollaries of the obtained results.

Social networks are represented using graph theory. In this case, individuals in a social network are assumed as nodes. Sometimes institutions or groups are also assumed as nodes. Institutions and such groups are assumed as cluster nodes that contain individuals or simple nodes. Hypergraphs have hyperedges that include more than one node. In this study, cluster hypergraphs are introduced to generalize the concept of hypergraphs, where cluster nodes are allowed. Sometimes competitions in the real world are done as groups. Cluster hypergraphs are used to represent such kinds of competitions. Competition cluster hypergraphs of semidirected graphs (a special type of mixed graphs called semidirected graphs, where the directed and undirected edges both are allowed) are introduced, and related properties are discussed. To define competition cluster hypergraphs, a few properties of semidirected graphs are established. Some associated terms on semidirected graphs are studied. At last, a numerical application is illustrated.

The different distance-based parameters are used to study the problems in various fields of computer science and chemistry such as pattern recognition, image processing, integer programming, navigation, drug discovery, and formation of different chemical compounds. In particular, distance among the nodes (vertices) of the networks plays a supreme role to study structural properties of networks such as connectivity, robustness, completeness, complexity, and clustering. Metric dimension is used to find the locations of machines with respect to minimum utilization of time, lesser number of the utilized nodes as places of the objects, and shortest distance among destinations. In this paper, lower bound of local fractional metric dimension for the connected networks is improved from unity and expressed in terms of ratio obtained by the cardinalities of the under-study network and the local resolving neighbourhood with maximum order for some edges of network. In the same context, the LFMDs of prism-related networks such as circular diagonal ladder, antiprism, triangular winged prism, and sun flower networks are computed with the help of obtained criteria. At the end, the bounded- and unboundedness of the obtained results is also shown numerically.

The importance of network security has increased with the emergence of networked systems in contemporary computing, making it an essential aspect of protecting digital assets and safeguarding against cyber threats. The current security mechanisms, which rely on cryptographic keys, may be susceptible to a number of attacks, such as media access control (MAC) spoofing, which might provide unauthorized users access to network resources. This study introduces a new approach, namely a zero-trust security framework with quantum fingerprinting for devices accessing a network, that utilizes quantum technology to protect networks from security threats and intruders. The proposed architecture relies on quantum fingerprinting to authenticate devices trying to access the network, and it is built on the zero-trust security concept. The framework is intended to offer a thorough, multi-layered approach to network security that may change in response to evolving security risks and specifications. By protecting against MAC spoofing and other types of device impersonation, the adoption of quantum fingerprinting adds another degree of protection. The proposed framework may be used to construct a reliable and scalable network security solution in different network environments.