Fuzzy autocatalytic set of fuzzy graph type-3 based on functional analysis theory

Abstract

Fuzzy Autocatalytic Set (FACS) of fuzzy graph Type-3 is one of the newly capable graphs for a breakthrough of the relationship between fuzzy graph and autocatalytic set. It was successfully used to model an incineration process of a regional clinical waste in 2005. In this research, the mathematical structures of FACS and its properties were explored based on the suitable concept of fuzzy and non-fuzzy metric structure and fuzzy and non-fuzzy normed structure with the structure of its graph. In other words, the possible new structures of FACS have been explored via two new insights of its metric fuzziness and its normed fuzziness. Firstly, a fuzzy detour FT3-distance between vertices in FACS was investigated whereby a quasi-pseudo-FT3-metric fuzziness on FACS which depends on this distance was established, followed by an introduction to the concept FT3-cycle space of FACS as a vector space which is then proven as a normed space. These concepts have led to the visualization of the structure of FACS in a fuzzy norm, hence some propositions and theorems were established. In addition, the study on these structures of FACS was exploited, in particular the connection with functional analysis features and the application of these structures to the clinical incineration process.