Sifat Fundamental Pada Granum Eulerian

Abstract

Mathematical analysis has several important connections with graph theory. Although initially, they may seem like two separate branches of mathematics, there are relationship between them in several aspects, such as graphs as mathematical objects that can be analyzed using concepts from analytic mathematics. In graph theory, one often studies distance, connectivity, and paths within a graph. These can be further analyzed using analytic mathematics, such as in the structure of natural numbers. Literature studies on graph theory, especially Eulerian graphs, are interesting to explore. An Eulerian path in a graph G is a path that includes every edge of graph G exactly once. An Eulerian path is called closed if it starts and ends at the same vertex. The concept of granum theory as a generalization of undirected graphs on number structures provides a rigorous approach to graph theory and demonstrates some fundamental properties of undirected graph generalization. The focus of this study is to introduce the connectivity properties of Eulerian granum. The granum G(e,M) is called connected if for every u,v ∈ M with u ≠ v there exists a path subgranumG^' (e,M^' )⊆ G(e,M)  where u,v ∈ M^' and is called an Eulerian granum if there exists a surjective mapping ϕ∶ [∥E(G(e,M))∥ + 1]→ M such that e(ϕ(n),ϕ(n+1))=1 for every n ∈ [‖E(G(e,M))‖]. This property provides a deeper understanding of the structure and characteristics of Eulerian granum, which have not been fully comprehended until now.