Research On The Operations And Properties Of Several Fuzzy Graphs

Fuzzy mathematics was born because of the study of uncertainty.So far,it has been highly valued by all aspects,and has made many excellent achievements in mathematical theory,application methods and practical applications.As a generalization of sets,fuzzy sets have penetrated into almost all branches of theoretical mathematics based on set theory,especially it is extremely active to study graph theory by using the idea of fuzzy sets.In 1975,Rosenfeld proposed the concept of fuzzy graphs,obtained a series of conclusions parallel to the graphs,and initially established the fuzzy graph theory.After that,many scholars have carried out in-depth and meticulous research on fuzzy graphs,and the fuzzy graph theory is developing rapidly.In view of the cognitive differences and thinking complexity of decision-makers,which lead to the hesitation of decision-makers,Torra proposed the concept of hesitation fuzzy sets in 2009.In order to study the fuzzy graph theory in the hesitant fuzzy environment,domestic scholars Chao Zhang and Deyu Li proposed the definition of hesitant fuzzy graphs in 2017,which has attracted much attention in multi-attribute decision-making problems.Based on the existing operations and related knowledge of fuzzy graphs,in this paper,the operations and related properties of several fuzzy graphs are studied,and mainly does the following work:Firstly,the intersection,union,complement,Cartesian product,direct product,strong product and lexicographic product of three kinds of fuzzy graphs,namely regular fuzzy graphs,bipartite fuzzy graphs and connected fuzzy graphs are defined.Also a series of properties that three kinds of fuzzy graphs satisfy the closure of the above operations under conditional constraints are studied.Secondly,combined with the concept of strong fuzzy graphs,the definitions of strong regular fuzzy graphs,strong bipartite fuzzy graphs and strong connected fuzzy graphs are given,and the intersection,union,complement,Cartesian product and direct product,strong product and lexicographic product of three kinds of strong fuzzy graphs are defined;and a series of properties of three kinds of strong fuzzy graphs satisfying the closure of the above operations under conditional constraints are studied.Finally,four operations and related properties of hesitant fuzzy graphs are supplemented;and a special hesitant fuzzy graph,namely one-step hesitant fuzzy graphs are studied.The definitions of one-step hesitant fuzzy graphs and strong one-step hesitant fuzzy graphs,complete hesitant fuzzy graphs are given.The closure and decomposition properties of strong one-step hesitant fuzzy graphs under Cartesian product,direct product,strong product,lexicographic product,and composition operations are studied.