In this paper, an unprecedented kind of fuzzy graph designated as m-polar interval valued fuzzy graph (m-PIVFG) is defined. Complement of the m-PIVFG open and closed neighborhood degrees of m-PIVFG are discussed. The other algebraic properties such as density, regularity, irregularity of the m-PIVFG are investigated. Moreover, some basic results on regularity and irregularity of m-PIVFG are proved. Free nodes and busy nodes of m-PIVFG is explored with some basic theorems and examples. Lastly, an application of m-PIVFG is described.

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In this paper, the concept of the m-polar fuzzy graph (m-PFG) and interval-valued fuzzy graph (IVFG) is integrated and introduced an unprecedented kind of fuzzy graph designated as m-polar intervalvalued fuzzy graph (m-PIVFG). Complement of the m-PIVFG is defined and the failure of this definition in some cases are highlighted. Various examples are cited and then redefined the notation of complement such that it applies to all m-PIVFGs. The other algebraic properties such as isomorphism, weak isomorphism, co-weak isomorphism of the m-PIVFG are investigated. Moreover, some basic results on the isomorphic property of m-PIVFG are proved. Finally, an application of m-PIVFG is explored.

The concept of domination is one of the most significant topics in graph theory to handle unpredictable phenomena. In this study, an unprecedented idea of domination is introduced in m - polar interval-valued fuzzy graph ( m -PIVFG). Domination number (DN), isolated vertex, total dominating set, independent set of domination on m -PIVFG are discussed. Some algebraic properties of domination on m -PIVFG are investigated. Weak domination, strong domination, split and non-split domination, cototal and global dominating sets on m -PIVFG are investigated with some fundamental hypotheses and models. We explore the concept of domination in m -PIVFG by solving a case study of locating new facilities to handle a catastrophe reaction activity due to the “COVID-19 pandemic” in West Bengal, India. Ultimately, conclusions and avenues of future scopes are placed at the end of this study.

A new concept of vertices in a fuzzy graph known as defective vertices is introduced here. A vertex in a fuzzy network is called defective if no edges incident with it are strong. Defective vertex cannot be ignored when determining dominance in a fuzzy network because they are a part of the network. Finding defective vertices in a network is not much difficult when the adjacency matrix is given. In this paper, the novel concept of defective vertices of a fuzzy graph is introduced. Based on this idea a stable domination set and a stable domination number are defined. This also optimised the network by establishing minimal connectivity. We have proposed three algorithms for finding the defective vertices, establishing stable connectivity, and determining the stable domination number for a given graph. An application of stable domination in the diagnosis of chickenpox disease is demonstrated to show the effectiveness of the proposed algorithms.