Fuzzy graph idea has turn out to be a dominant idea for modeling and fixing combinatorial optimization issues that come up in several fields. The idea of fuzzy graphs was narrated by Rosenfeld , on the premise of Zadehs fuzzy units . The creator has obtained the fuzzy analogues of a number of primary graph theoretical ideas. Bhutani  launched the idea of automorphisms of fuzzy graphs and outlined an entire fuzzy graph. Mordeson and Nair  have established a ample and mandatory situation for a fuzzy graph which is cycle to be a fuzzy cycle.
Sunitha and Vijayakumar  have characterised fuzzy bushes utilizing its distinctive most spanning tree. The sturdy arcs in fuzzy graph have been launched by Bhutani and Rosenfeld . The authors have additionally studied on the sturdy arcs of a fuzzy tree. Relying on the energy of an arc, Mathew and Sunitha  have categorized arcs in fuzzy graph into sturdy (?-strong and ?-strong) and never sturdy arcs (?-arcs).
Fuzzy coloring of fuzzy graphs performs an necessary position in fixing community issues.
Munoz et. al  launched the idea of coloring of fuzzy graphs. Later, Eslahchi and Onagh  launched fuzzy coloring of fuzzy graph and outlined fuzzy chromatic variety of a fuzzy graph. A number of authors together with Kishore and Sunitha  and Samanta et. al  have labored on the fuzzy coloring of fuzzy graphs. In , Kishore and Sunitha have initiated the idea of sturdy coloring of fuzzy graphs based mostly on sturdy arcs and outlined sturdy chromatic variety of a fuzzy graph. In , they’ve additionally studied on sturdy chromatic variety of resultant of fuzzy graphs. Recently, Mamo and Srinivasa Rao  launched the idea of fuzzy chromatic polynomial of fuzzy graph based mostly on ?-cuts of the fuzzy graph that are crisp graphs. On this analysis article, we introduce sturdy chromatic polynomial in fuzzy graph, known as sturdy fuzzy chromatic polynomial of fuzzy graph. We research the sturdy fuzzy chromatic polynomial of some fuzzy graph buildings. Additional, we receive the relation between sturdy fuzzy chromatic polynomial and fuzzy chromatic polynomial for some fuzzy graph buildings.
The group of this analysis article is as follows. In Part 2, we evaluate some primary ideas on fuzzy graphs, varieties of arcs in fuzzy graph and robust coloring of fuzzy graph. In Part three, we outline sturdy fuzzy chromatic polynomial of fuzzy graph based mostly on sturdy coloring of fuzzy graph. Additionally, we established a ample and mandatory situation for sturdy fuzzy chromatic polynomial of fuzzy graph to be the chromatic polynomial of its underlying crisp graph. From Part Four-7, we current the sturdy fuzzy chromatic polynomial of some fuzzy graph buildings. In Part eight, we current conclusion.