الفهرس 
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Abstract
The science of mathematics have been expanded widely in the last years. it contains now different well established sciences by it self. like statistic, probability theory, stochastic processes, theoretical computer science and operations research to name a few.
In this thesis we combine some particular concepts of different branches of mathematics. markov chains from the stochastic process and operations research, formal graph theory from the theoretical computer science and the eigen problem from the linear algebra see mital and mohan and james, r.schoot.
This thesis deals with the problem of finding a representation of any markov chain in a new form, as a mathematical structure, namely its graph theory representation.
The primary focus of the thesis is on problems in which we define the graph in a new form, as a mathematical structure, also in which we define a markov chain and call it the original markov chain and hence we present it in a graph. also we define a new function on this markov chain called an expanded markov chain in teo types, the first is called the first expand maekov chain in which our main new problem is to represent it in a new form(graph theory representation). our main problem, in this case, is to find its eign values( or its characteristic roots) in order to find its eigen(or its characteristic) vector space which in turns help us in analyzing the original markov chain and its first expanded markov chain . the second type is called the second expanded markov chain in which we aim also to define it in define it in a new formal definition and repersent it in the new form (gragh theory representation).