It is much more practical to use the [Formula: see text]-norm than the minimum in defining the fuzzy graph. Hence, the concept of [Formula: see text]-fuzzy graph was introduced by Mordeson et al. in [J. N. Mordeson and S. Mathew, [Formula: see text]-norm Fuzzy Graphs, New Mathematics and Natural Computation 14(1) (2018) 129–143]. In this paper, we concentrate on the relatively new concept of fuzzy adjacency matrix of that [Formula: see text]-fuzzy graphs. The purpose of this paper is to further develop a product of this fuzzy adjacency matrix based on [Formula: see text]-fuzzy graphs. Moreover, two kinds of Boolean matrix are defined and relation between Normal Boolean matrix, product of adjacency matrix and 1-level cut Boolean matrix is obtained.
In this paper, we investigate the fuzzification of zero forcing process. For this, first we introduce a new embedding of a graph [Formula: see text] by considering a minimal zero forcing set of [Formula: see text] and an arbitrary list of maximal forcing chains of this zero forcing set. Then we get a comparison between zero forcing sets of a graph by using fuzzy concepts. Finally, we give an application for this procedure.
Graph models are found everywhere in natural and human made structures, including process dynamics in physical, biological and social systems. The product of graphs are appropriately used in several combinatorial applications and in the formation of different structural models. In this paper, we present a new product of graphs, namely, maximal product of two vague graphs. Then we describe certain concepts, including strongly, completely, regularity and connectedness on a maximal product of vague graphs. Further, we consider some results of edge regular and totally edge regular in a maximal product of vague graphs. Finally, we present an application for optimization of the biomass based on a maximal product of vague graphs.
Fuzzy graph theory is finding an increasing number of application in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Special fuzzy graph can be obtained from two given fuzzy graphs using the operations beta products. In this paper, we introduce the notions of some kinds of beta product of two fuzzy graphs. The concept of strong, regular and complement of [Formula: see text]-product of two fuzzy graphs and relation between them are also obtained. At the end, an application with a cryptographic object is said to be using the [Formula: see text]-product of fuzzy graphs.
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