Jul 2026· New Mathematics and Natural Computation· pp. 1-36· 0 citations
TL;DR
The novel concept of Steiner geodesic distance (SGD) in fuzzy graphs is introduced, which serves as a broader extension of the geodesic distance, and applications in optimizing fiber optic cable routing and identifying the optimal location for a central command hub in encrypted radio communication networks are presented.
Abstract
Distance in fuzzy graphs is a fundamental concept crucial in analyzing connectivity patterns and network dynamics. This article aims to extend and generalize the notion of distance beyond pair of vertices by introducing the novel concept of Steiner geodesic distance (SGD) in fuzzy graphs, which serves as a broader extension of the geodesic distance. If a fuzzy graph has n vertices, this approach provides a framework for determining the distance among k vertices, where [Formula: see text], thereby laying the foundation for analyzing interactions among multiple vertices in the network. The Steiner geodesic tree is defined and the bounds of SGD are obtained. An algorithm for finding Steiner geodesic (SG) in [Formula: see text] running time is presented. The notions of SG k - eccentricity, SG k - center, SG k - selfcentered fuzzy graphs and SG k - eccentric set are established and a characterization of SG k - selfcentered fuzzy graphs is provided. The SGD concepts are analyzed on fuzzy graphs including complete fuzzy graph, complete bipartite fuzzy graph, fuzzy cycle and fuzzy trees. The idea of SGD k - matrix is also discussed and proposed an algorithm to distinguish between SG k - selfcentered fuzzy graphs. The article also presents applications of SGD concepts in optimizing fiber optic cable routing and identifying the optimal location for a central command hub in encrypted radio communication networks.
Computer science applications make extensive use of graph theory. Information mining, picture classification, grouping images, photo capture, and connectivity are particularly important study areas in computer science. Fuzzy labelling models provide better accuracy, adaptability, and interoperability to the framework than conventional fuzzy models. They are used in many areas of basic mathematics, including traditional computer science and physical scientific study. A common problem in mathematics is the distance properties in a graph. In fuzzy mean labelling graphs, paths have some exciting applications. Using lengths to securely identify the vertex in an organization is one such application. As the outcome, in this study, we explore four paths that are DSMFLs in graphs: comb graph, double comb, path-attached pendant vertex, and path-attached two pendant vertices. In fuzzy labelling graphs, there exist several DSMFLs. Whenever the force of connection between each pair of vertices G matches the value of the membership of the edges, G appears self-sufficient in relation to the DSMF. Furthermore, it is established that any interconnected fuzzy labelling graph is both a path graph and a comb graph.
Uma Maheswari Subbiah, Shalini Ramachandran· Advanced International Journ...· 0 citations
This paper introduces and investigates the connectivity properties with [Formula: see text]-PFIG, a novel extension of fuzzy incidence graphs characterized by multiple membership values corresponding to different polarities. The study systematically defines and explores key connectivity concepts, including paths, cycles, and connectedness, under the [Formula: see text]-polar fuzzy framework. The authors present theoretical findings that expand traditional graph connectivity concepts to a more generalized context, offering a deeper understanding of the behavior and structure of [Formula: see text]-PFIG. This study introduces an innovative method for connectivity concept and incidence graph within [Formula: see text]-polar fuzzy graphs ([Formula: see text]-PFG). Several properties, along with detailed descriptions are thoroughly investigated. The proposed approach accounts for the various merits of each member in a railway network system by considering individual merits (self-weight) in the nature.
Dilara Akter Mitu, Tanmoy Mahapatra, Abid Ali et al.· Journal of Interconnection N...· 0 citations
The ‘divide and conquer’ paradigm proves to be one of the most frequently used techniques for dealing with the complexities of graph-related problems. Therefore, it is of great importance to measure the tendency of a vertex to be critical and its susceptibility in a graph. The criticality of a vertex is often analysed in terms of its strength. Removing a highly critical vertex from a graph modelling a network may introduce vulnerability into the system represented by the graph. Minimizing the vulnerability of such a network without affecting its fundamental structure, thereby improving the stability of the graph, is the primary objective of the article. To achieve this, certain properties of Euler graphs are analysed in terms of vertex strength, and a method is presented for determining all possible constructions of Euler graphs corresponding to different integer partitions. The parts of a partition represent the vertex strengths, and their sum corresponds to the total vertex strength of the graph. Various connectivity indices are employed to validate the proposed constructions. Furthermore, their interrelationships and potential real-life applications are also discussed. It is evident from the constructions that they may play a vital role in developing network deception technology to protect digital assets, as each partition of the network generates a distinct network.
Saifur Rahman, Raju Doley· Acta Universitatis Sapientia...· 0 citations
This paper introduces, for the first time, a novel structure called the permanent graph (permanent fuzzy graph), utilizing the established concepts of neighborhood and distance between two nodes, and proposes a method to revive a graph whose functionality has been compromised due to the loss of nodes, thereby preventing its complete collapse.
E. Darabian, R. Borzooei· Soft Computing - A Fusion of...· 0 citations
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.
Zeeshan Saleem Mufti, A. H. Tedjani, Shama Liaqat et al.· Scientific Reports· 0 citations
Let G be a connected simple graph with the vertex set V(G)={v1,v2,…,vn}, where the distance dG(vi,vj) is the length of a shortest path between vi and vj, and the distance matrix D(G)=(dij)n×n is defined by dij=dG(vi,vj). Being real symmetric and non-negative, D(G) has real eigenvalues λ1(G)≥λ2(G)≥…≥λn(G), with λn(G) referred to as the least distance eigenvalue of G. In this paper, we give the maximum value of the least distance eigenvalue, and determine the unique extremal graph with two pendant vertices whose complements attain the value.
Jinfeng Zhang, Xu Chen, Jia-Wen Deng et al.· Mathematics· 0 citations
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