Degree, Neighbourhood, and Connectedness in Fuzzy Digraph

Objective: In the context of fuzzy digraphs, degree neighborhood, and connectedness play important roles in understanding the structure and relationships within the graph. Here are the objectives of degree, neighborhood, and connectedness in fuzzy digraphs: 1. Degree Neighborhood: The degree neighborhood of a vertex in a fuzzy digraph refers to the set of vertices that are directly connected to it. In fuzzy digraphs, the concept of degree neighborhood is extended to include the strength or degree of connection between vertices, which is represented by fuzzy values. 2. Connectedness: Connectedness in fuzzy digraphs refers to the existence of paths between any pair of vertices, considering the fuzzy nature of relationships. Methods: By using fuzzy vertex and fuzzy edges in fuzzy digraph to find degree (in-degree and out-degree), neighborhood (in-neighborhood and out-neighborhood), and to find connectedness in fuzzy digraph. Findings: In this study, some types of fuzzy digraphs are introduced, neighborhood (in-neighborhood and out-neighborhood) and degree (in-degree and out-degree) in fuzzy digraphs, and strong, weak, unilateral connectedness in fuzzy digraphs are introduced. Novelty: To show that let 𝐺𝐷 ∶ (𝜎, 𝜇) be a fuzzy digraph with fuzzy vertex set 𝜎 and fuzzy edge set 𝜇. In this paper, some types of fuzzy digraphs are introduced, neighborhood (inneighborhood and out-neighborhood) and degree (in-degree and out-degree) in fuzzy digraph and strong, weak, unilateral connectedness in fuzzy digraph are introduced. Furthermore, this paper establishes various results and theorems, and some properties of different fuzzy digraphs are discussed. Keywords: Fuzzy digraph; Degree in a fuzzy digraph; Neighborhood in a fuzzy digraph; Fuzzy diwalk; Fuzzy dicycle; Fuzzy diacyclic digraph