الفهرس 
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Abstract
This thesis proposes to study novel concepts of fuzzy graph, new concepts of soft sets and rough sets and link them to the medical concepts. Novel approaches are introduced for creating various kinds of fuzzy subgraphs described as K ̃_1 K ̃_2- fuzzy subgraphs depending on the variable memberships of the vertices and edges, as well as defining a new type of path P^(κ_1 κ_2 ). The technique used focused on investigating how strength affects various types of edges through the explanation of numerous examples and the inference of new significant features. Also, the hepatic portal system is represented as a graph, and all of the above using the hepatic portal system circulation is illustrated by linking the fuzzy graph to the medical field. In addition, the influence of the resultant properties on medical occurrences will be analyzed, as well as perspectives on the repercussions of the consequences of abnormal cases. A new algorithm is used by matlab methods in order to calculate the membership of vertices through the effect of each vertex in the hepatic portal system, as well as the membership of edges via the effect of each edge in the hepatic circulation.Applying several kinds of fuzzy graph operations through the concept of K ̃_1 K ̃_2 - fuzzy subgraph using different methods that based on the changeable of K ̃_(1 ) and K ̃_2 of the fuzzy graph. The operations include (U ̃_1 U ̃_2)n -union of fuzzy subgraphs, (I ̃_1 I ̃_2)n intersection of fuzzy subgraphs, (J ̃_1 J ̃_2)n -join of fuzzy subgraphs, (C ̃_1 C ̃_2 )_nand (N ̃_1 N ̃_2 )_n -complements of fuzzy subgraphs, (P ̃_1 P ̃_2)n cartesian product of fuzzy subgraphs , and (M ̃_1 M ̃_2)n composition of fuzzy subgraphs. Many characteristics are studied by using various approaches under the influence of the strength of the fuzzy graph. And lastly, the comparison properties and containment relationships between various kinds of the resultant fuzzy subgraphs will be discussed.Presenting the soft set applications in a decision-making problem for the healthcare sector that helps in reaching the best possible option for a situation or a problem with the aid of Pawlak’s rudimentary mathematics and an algorithm of choice value. They assist the specialist doctor in making the decision by classifying many patients after determining their medical diagnosis based on the various symptoms of viruses to take the necessary procedures and precautions. The problem is to decide which of the six patients has severe and advanced COVID-19 illness signs must be hospitalized or quarantined. It depends on the criteria specified by the specialist doctor for selection.Novel types of subgraphs are created according to attributes of the main graph through generalized relationships through the out-link vertices or directed cycles by using soft sets called cycled soft graphs that are denoted by ꞒⱾ -graph. The new concept of cycled soft graphs with vital and medical phenomena is strengthened by representing the cardiovascular system of the human body as a graph and interpreting each one of them from the standpoint of mathematics and medical explanations of occurrences and facts. New methods are presented for establishing the lower and upper approximations for both of the edges and the vertices of a portion of G*. It focuses on exploiting closed paths, namely that the vertices lie on closed paths, and using a set of parameters that connect a set of vertices to each other. A new type of rough soft graph is introduced that is denoted by the cycled rough soft graph, written as ꞒRⱾG^ς(G^* ). The cardiovascular system of the human body is used to apply all of these concepts mathematically in order to describe a variety of occurrences and medical facts. Additionally, it is discussed the containment relationships and the comparison properties associated with numerous new approximation soft subgraph types. Finally, several operations on rough soft graphs are introduced through the utilization of examples to illustrate the novel concept of approximation soft graphs.