In this writing, first, we disclose the first and second category of a ΓτF-fuzzy proximal contraction for a mapping O:U→V which is nonself and also declare a fuzzy q-property to confirm the existence of the best proximity point for nonself function O. Then, we discover a few results using the ΓτF-fuzzy proximal contraction of the first category for a continuous and discontinuous nonself function O in a non-Archimedean fuzzy metric space. Later, we discuss another result for the ΓτF-fuzzy proximal contraction of the second category as well. In between the fuzzy proximal theorems, many examples are presented in support of the definitions and theorems proved in this writing.
Graphs whose spectrum belongs to the interval [−2, 2] are called Smith graphs. The structure of a Smith graph with a given spectrum depends on a system of Diophantine linear algebraic equations. We have established in [1] several properties of this system and showed how it can be simplified and effectively applied. In this way a spectral theory of Smith graphs has been outlined. In the present paper we introduce cospectrality graphs for Smith graphs and study their properties through examples and theoretical consideration. The new notion is used in proving theorems on cospectrality of Smith graphs. In this way one can avoid the use of the mentioned system of Diophantine linear algebraic equations.
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