Fuzzy Graph Theory

Fuzzy graph theory has become a dominant concept for modeling and solving combinatorial optimization problems that come up in different fields. The concept of fuzzy graphs was narrated by Rosenfeld [1], on the basis of Zadeh's fuzzy sets [2]. The author has obtained the fuzzy analogues of several basic graph theoretical concepts. Bhutani [3] introduced the concept of automorphisms of fuzzy graphs and defined a complete fuzzy graph. Mordeson and Nair [4] have established a sufficient and necessary condition for a fuzzy graph which is cycle to be a fuzzy cycle.

Sunitha and Vijayakumar [5] have characterized fuzzy trees using its unique maximum spanning tree. The strong arcs in fuzzy graph were introduced by Bhutani and Rosenfeld [6]. The authors have also studied on the strong arcs of a fuzzy tree. Depending on the strength of an arc, Mathew and Sunitha [7] have classified arcs in fuzzy graph into strong (?-strong and ?-strong) and not strong arcs (?-arcs).

Fuzzy coloring of fuzzy graphs plays an important role in solving network problems.

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Munoz et. al [8] introduced the concept of coloring of fuzzy graphs.

Later, Eslahchi and Onagh [9] introduced fuzzy coloring of fuzzy graph and defined fuzzy chromatic number of a fuzzy graph. Several authors including Kishore and Sunitha [10] and Samanta et. al [11] have worked on the fuzzy coloring of fuzzy graphs. In [12], Kishore and Sunitha have initiated the concept of strong coloring of fuzzy graphs based on strong arcs and defined strong chromatic number of a fuzzy graph. In [13], they have also studied on strong chromatic number of resultant of fuzzy graphs. Lately, Mamo and Srinivasa Rao [14] introduced the concept of fuzzy chromatic polynomial of fuzzy graph based on ?-cuts of the fuzzy graph which are crisp graphs. In this research article, we introduce strong chromatic polynomial in fuzzy graph, called strong fuzzy chromatic polynomial of fuzzy graph. We study the strong fuzzy chromatic polynomial of some fuzzy graph structures. Further, we obtain the relation between strong fuzzy chromatic polynomial and fuzzy chromatic polynomial for some fuzzy graph structures.

The organization of this research article is as follows. In Section 2, we review some basic concepts on fuzzy graphs, types of arcs in fuzzy graph and strong coloring of fuzzy graph. In Section 3, we define strong fuzzy chromatic polynomial of fuzzy graph based on strong coloring of fuzzy graph. Also, we established a sufficient and necessary condition for strong fuzzy chromatic polynomial of fuzzy graph to be the chromatic polynomial of its underlying crisp graph. From Section 4-7, we present the strong fuzzy chromatic polynomial of some fuzzy graph structures. In Section 8, we present conclusion.