latest omprovement in fuzzy graphs and open questions

fuzzy graph theory applications

ABSTRACT

Abstract. A fuzzy graph (f-graph) is a pair G : (s, μ) where s is a fuzzy subset of a set S and μ is a fuzzy relation on s. A fuzzy graph H : (t, u) is called a partial fuzzy subgraph of G : (s, μ) if t (u) £ s(u) for every u and u (u, v) £ μ(u, v) for every u and v . In particular we call a partial fuzzy subgraph H : (t, u) a fuzzy subgraph of G : (s, μ ) if t (u) = s(u) for every u in t * and u (u, v) = μ(u, v) for every arc (u, v) in u*. A connected f-graph G : (s, μ) is a fuzzy tree(f-tree) if it has a fuzzy spanning subgraph F : (s, u), which is a tree, where for all arcs (x, y) not in F there exists a path from x to y in F whose strength is more than μ(x, y). A path P of length n is a sequence of distinct nodes u0, u1, ..., un such that μ(ui−1, ui) > 0, i = 1, 2, ..., n and the degree of membership of a weakest arc is defined as its strength. If u0 = un and n³ 3, then P is called a cycle and a cycle P is called a fuzzy cycle(f-cycle) if it contains more than one weakest arc . The strength of connectedness between two nodes x and y is defined as the maximum of the strengths of all paths between x and y and is denoted by CONNG(x, y). An x − y path P is called a strongest x − y path if its strength equals CONNG(x, y). An f-graph G : (s, μ) is connected if for every x,y in s* ,CONNG(x, y) > 0. In this paper, we offer a survey of selected recent results on fuzzy graphs.