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The isometry in crisp graph theory is a well-known fact. But, isometry under a fuzzy environment was developed recently and studied many facts. In a m-polar fuzzy graph, we have to think m components for each node and edge. Since, in our consideration, we consider m components for each nodes as well as edges, therefore we can not handle this type of situation using fuzzy model as their is a single components for this concept. Again, we can not apply bipolar or intuitionistic fuzzy graph model as each edges or nodes have just two components. Thus, these mPFG models give more efficient fuzziness results than other fuzzy model. Also, it is very interesting to develop and analyze such types of mPFGs with examples and related theorems. Considering all those things together, we have presented isometry under a m-polar fuzzy environment. In this paper, we have discussed the isometric m-polar fuzzy graph along with many exciting facts about it. Metric space properties have also been implemented on m-polar fuzzy isometric graph. We also have initiated a generalized fuzzy graph, namely antipodal m-polar fuzzy graphs, along with several issues. The degree of it is also presented along with edge regularity properties. We also give a relation between m-polar fuzzy antipodal graphs and their underlying crisp graphs. Its properties have also been discussed on m-polar fuzzy odd as well as even cycles, complete graphs, etc. Finally, a real-life application on a road network system in a m-polar fuzzy environment using the $$\mu$$ μ -distance concept is also presented.

The tolerance graph is a well-defined topic for a crisp graph theory. But, tolerance in a fuzzy graph is defined recently and many properties have been investigated. But, in a fuzzy graph, only one tolerance is considered for every vertex. But, in an m-polar fuzzy graph (mPFG), each vertex and edge has a m number of membership values. So, defining tolerance for mPFG is not easy and needs some new ideas. By considering m tolerances for every component out of m components of membership values of a vertex or edge, we defined m-polar fuzzy tolerance graph (mPFTG). Here, m-polar fuzzy convex set (mPFCS) as well as strongly mPFCS are also studied. Some different types of mPFTGs like m-polar fuzzy min-tolerance graph, m-polar fuzzy max-tolerance graph, m-polar fuzzy sum-tolerance graph are also introduced. Some related terms like bounded mPFTG, m-polar fuzzy interval containment graph, m-polar fuzzy unit tolerance graph (mPF unit TG) are also defined. Here, m-polar fuzzy intersection graph along with tolerance core as well as tolerance support length are also discussed. Some interesting properties on m-polar fuzzy min-tolerance graph, m-polar fuzzy max-tolerance graph, m-polar fuzzy sum-tolerance graph are also investigated. Lastly, a real-life application based on an assigned work done by a private company has been discussed to show its practicability in mPFTG.

The threshold graph is a well studied topic. But, the fuzzy threshold graph is defined recently and investigated many properties. In a fuzzy threshold graph, only one threshold is considered for every vertex and edge. In a m-polar fuzzy graph, each vertex and each edge has a m number of membership values. So, defining a threshold graph for m-polar fuzzy graph is not easy and needs some new ideas. By considering m thresholds each for each component out of m components of membership values of a vertex or edge, we defined m-polar fuzzy threshold graph (mPFTG). Many interesting properties are also presented on mPFTG. The mPFTG is also decomposed in a very unique way and this decomposition generates m different fuzzy threshold graphs and the properties of decomposed graphs are explained. The important parameters viz. m-polar fuzzy threshold dimension and mPF threshold partition number are also defined and investigated thoroughly. Finally, a real-life application using mPFTG on a resource power controlling system is presented.

In this article, a new idea of interval-valued m-polar fuzzy (IVmPF) graph is introduced and investigated some of it’s properties. Here, IVmPF multiset, interval-valued m-polar fuzzy (IVmPF) multi graph are presented. IVmPF planar value of an interval-valued m-polar fuzzy (IVmPF) planar graph along with degree of planarity value is also introduced to measure the planarity value of an interval-valued m-polar fuzzy (IVmPF) graph. Some related terms like complete IVmPF graph, strong IVmPF graph, strong edges, faces of IVmPF planar graph are presented. In this paper, IVmPF dual graph is also described which is closely related to IVmPF planar graph. Some important properties are studied on IVmPF dual graph. Lastly, a real life application on IVmPF planar graph has been discussed to show its practicability.

In this article, a new idea of fuzzy fractional coloring of fuzzy graph is presented and fuzzy fractional chromatic number is defined. A relationship between fuzzy fractional chromatic number and fuzzy fractional clique number is established. Some properties of fuzzy chromatic number of fuzzy graphs and fuzzy fractional chromatic number of fuzzy graphs are proved and the concept of k-strong adjacent vertices is introduced. Fuzzy chromatic number and fuzzy fractional chromatic number have been calculated on lexicographic product of two fuzzy graphs. Also, fuzzy chromatic number, independence number and fuzzy fractional chromatic number have been investigated on disjoint union of two fuzzy graphs. Lastly, a real life application of fuzzy fractional coloring on fuzzy graph is discussed.