Abstract
This paper states a new concept of highly \(D_2\)-distance of irregular labeling fuzzy graphs and highly totally \(D_2\) -distance of irregular labeling fuzzy graphs with a path on four vertices, Barbell graph and a cycle of length \(\geqslant 4\). Some properties related to neighbourly totally \(D_2\)-distance of irregular labeling fuzzy graphs, highly totally \(D_2\) -distance of irregular labeling fuzzy graphs and product fuzzy graphs are discussed with some special graphs. In addition, some more properties and examples of these graphs are studied. The highly \(D_2\)-distance of irregular labeling fuzzy graphs and neighbourly \(D_2\)-distance of irregular labeling fuzzy graphs are observed and viewed for highly totally \(D_2\)-distance of irregular labeling fuzzy graphs and neighbourly totally \(D_2\)-distance of irregular labeling fuzzy graphs.
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