Graphics and representable fuzzifying matroids

Types of arcs in a fuzzy graph

Strong arcs in fuzzy graphs

Median graphs represent a unique category of graphs with significant importance in distance-related concepts within graph theory. This paper aims to expand the concept of median graphs into fuzzy graph theory and investigate whether every fuzzy tree belongs to the classification of the fuzzy median graph. Furthermore, this study characterizes the strong edges in a fuzzy graph using median sets. Additionally, the work delves into the impact of removing bridges and cutvertices on a fuzzy median graph. Finally, the paper proposes a novel model rooted in the concept of the fuzzy median graph for deploying wireless sensors, which is meticulously compared with the widely used branch and bound algorithm. The proposed model demonstrates superior performance in significantly reducing the inter-sensor distances.

In this paper, the notion of cycle connectivity of bipolar fuzzy graphs is introduced. The concept of strong edge, $P$ cut node, $P$-bridge and $\theta$ - evaluation of bipolar fuzzy graphs are studied. The cycle connectivity of bipolar fuzzy trees, bipolar fuzzy cycles and complete bipolar fuzzy graphs are obtained. A condition for complete bipolar fuzzy graphs to have cyclic cut-nodes is obtained.

The concept of domination is introduced within the context of signed fuzzy graphs (signed-FGs), along with examples, as a novel metric to evaluate graph stability under varying conditions. This metric particularly focuses on dominant sets and integrity measures, providing a well-rounded approach to assessing the structural stability of signed- FGs. The necessity of fulfilling the domination integrity condition in evaluating the performance of signed-FGs is highlighted through a discussion on its formulation and an analysis of its upper and lower bounds. An algorithm for identifying strong arcs and classifying them is presented, along with an algorithm for identifying signed fuzzy trees. Furthermore, the role of symmetry in signed-FGs is explored, revealing that symmetrical structures often correspond to higher domination integrity, thus contributing to the improved stability and predictability of the graphs. The paper also establishes important connections with classical graph varieties, such as complete graphs and their variations, demonstrating that changes in domination integrity increase with certain parameters. Additionally, real-life scenarios where these concepts are applicable serve to complement the theoretical results. The case study findings illustrate the significance of domination integrity in practical contexts by emphasizing various instances where it can be determined and utilized. Such instances include identifying independent dominant sets in path and cycle diagrams, as well as estimating the lower bounds of domination integrity in these structures. The estimation of domination integrity using block graph methods is underscored as crucial for enhancing the efficiency of signed-FG applications.

A fuzzy graphic matroid can be represented over a finite field F. The neccesary and sufficient conditions of a fuzzy graphic matroid to be binary are given. Also, discussed the fuzzy graph-theoretic context in which fuzzy matroids arise. Fuzzy matroid operations, which will illuminate more analogies that corresponds to the operations in fuzzy graph theory and matrix theory. The traditional network models stores the received packets and forward them without applying any additional process to the packets. The uncertainties caused during the transmission of messages in the traditional network models are rectified and noiseless trans- mission of messages in the network model is improved. The primary emphasis of this paper lies in the exploration of network codes and their correlation with fuzzy graphic matroid.

The theory of q -rung orthopair fuzzy sets ( q -ROFSs) is emerging for the provision of more comprehensive and useful information in comparison to their counterparts like intuitionistic and Pythagorean fuzzy sets, especially when responding to the models of vague data with membership and non-membership grades of elements. In this study, a significant generalized model q -ROFS is used to introduce the concept of q -rung orthopair fuzzy vector spaces ( q -ROFVSs) and illustrated by an example. We further elaborate the q -rung orthopair fuzzy linearly independent vectors. The study also involves the results regarding q -rung orthopair fuzzy basis and dimensions of q -ROFVSs. The main focus of this study is to define the concepts of q -rung orthopair fuzzy matroids ( q -ROFMs) and apply them to explore the characteristics of their basis, dimensions, and rank function. Ultimately, to show the significance of our proposed work, we combine these ideas and offer an application. We provide an algorithm to solve the numerical problems related to human flow between particular regions to ensure the increased government response action against frequently used path (heavy path) for the countries involved via directed q -rung orthopair fuzzy graph ( q -ROFG). At last, a comparative study of the proposed work with the existing theory of Pythagorean fuzzy matroids is also presented.