Cycles and cocycles of fuzzy graphs

In this paper we show that if the fuzzy graph (σ,μ) is a cycle, then it is a fuzzy cycle if and only if (σ,μ) is not a fuzzy tree. We also examine the relationship between fuzzy bridges and cycles. We introduce and examine the concepts of chords, twigs, 1-chains with boundary zero, cycle vectors, coboundary, and cocycles for fuzzy graphs. We show that although the set of cycle vectors, fuzzy cycle vectors, cocycles, and fuzzy cocycles do not necessarily form vector spaces over the field Z 2 of integers modulo 2, they nearly do. Thisallows us to introduce the concepts of (fuzzy) cycle rank and (fuzzy) cocycle rank for fuzzy graphs in a meaningful way.