Cardinality and Isomorphic Properties of Hajós Graphs and Hajós Fuzzy Graphs

Objectives: Fuzzy graphs allow uncertainty in the ideas characterizing vertices and edges to be included when modelling real-world scenarios into graph models. This study aims to introduce Hajós fuzzy graph, cardinality of Hajós graph and Hajós fuzzy graph and to explore many properties. Methods : Here, the Hajós construction is applied to two fuzzy graphs, and the Hajós fuzzy graph is defined by assigning membership values to all the edges and vertices of the newly formed fuzzy graph. Fuzzy graph conditions are verified for the assigned membership values. Examples are used to illustrate the concept. Findings: The Hajós fuzzy graph's size and order are established. Isomorphic property is discussed. The number of Hajós (fuzzy) graphs that exist for some combinations of two (fuzzy) graphs is calculated. Novelty: Based on Hajós construction, the Hajós fuzzy graph is defined. Based on the fact that different choices of vertices and edges produce different Hajós (fuzzy) graph, its cardinality is defined. The notion of cardinality is a novel concept in Hajos graph which helps to find the total number of Hajós (fuzzy) graphs. The cardinality of the Hajós (fuzzy) graph which arises from two (fuzzy) graphs that can be a cycle, path, regular graph, complete graph and complete bipartite graph are derived. Mathematics subject classification. (2010): 05C72, 05C76 Keywords: Hajós Graph, Hajós fuzzy graph, Order, Size, Isomorphism, Cardinality of Hajós fuzzy graph