This manuscript is aimed to deal with the isomorphism properties of total fuzzy graph. In order to achieve the desired objective, we consider two fuzzy graphs and their total fuzzy graphs. For these total fuzzy graphs, the notion of homomorphism of total fuzzy graphs is defined and the existence of a homomorphism between these two total fuzzy graphs is shown by examples. This is achieved by both sketching their graph and verifying the condition of homomorphism. Having a mapping being a homomorphism of the total fuzzy graphs the concept of weak isomorphism and co-weak isomorphism are presented with separate and distinct definition. These definitions are accompanied with supportive examples and the case where a homomorphism between two total fuzzy graphs is not weak-isomorphism is observed and illustrated by supportive example. In addition to these, the notion of co-weak isomorphism is also defined and illustrated by using a typical example. A very great attention is given to the definition of an isomorphism of total fuzzy graph and results arising from this definition. Accordingly, certain theorems related to isomorphic properties of total fuzzy graphs are stated and each of these theorems are thoroughly proved to show the results. Finally the manuscript states the future research works, limitations of the manuscript.

Keywords: Fuzzy graph, Total fuzzy graph, Homomorphism, Isomorphism, Week isomorphism

The data used to in these findings are included within the manuscript.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.