Fuzzy topological analysis of fuzzy Helm graphs with applications to protein interaction networks

In biological and chemical networks, in which the strength of interactions is often not absolute, uncertainty is an inherent feature of many real-world networks. A fuzzy graph consists of vertices with fuzzy memberships and fuzzy relationships among vertices and edges; therefore, fuzzy graph theory is a natural approach for modelling systems in which the connections between the vertices are fuzzy or not precisely known. Helm graphs are constructed by adding pendant vertices to the outer cycle of a wheel graph and have a layered hierarchy similar to that found in hub-and-spoke networks, such as those of protein interaction and communication networks. Encouraged by this structural similarity, in the present study, some degree-based fuzzy topological indices are investigated on fuzzy Helms graphs. Specifically, we obtain closed-form analytical expressions for the fuzzy Zagreb, fuzzy Randić, fuzzy Harmonic, fuzzy F-index, fuzzy Sombor, fuzzy Misbalance Prodeg, and fuzzy Nirmala indices. The formulas are derived directly from the partition of the type of vertices of the fuzzy Helm graph and are checked symbolically and by a numerical example. The deduced expressions show the dependence of these descriptors on structural properties, such as pendant attachments, edge weight distribution, and connectivity of the hubs. As a concrete application, the proposed framework is used in the case of the P53 protein interaction network, a central pathway in cancer biology. The numerical results illustrate the ability of fuzzy descriptors to differentiate between the dominant regulatory role of P53 and the role played by interacting partners, which cannot be achieved by classical crisp indices because they lose the graded aspect of biological interaction confidence. Therefore, this study suggests the use of fuzzy topological indices for network analysis in a wider context and suggests the natural development of the said approach to other families of fuzzy graphs, as well as to uncertain networks in chemical and biological applications.