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Abstract
We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These include: the number of connected functional graphs, the number of graphs having a maximal cycle, the number of cycles of fixed size, the number of components of fixed size, as well as the shape of trees extracted from functional graphs. We particularly focus on connected functional graphs, that is, the graphs that contain only one component (and thus only one cycle). Based on the results of our computations, we formulate several conjectures highlighting the similarities and differences between these functional graphs and random mappings.
Original language  English 

Pages (fromto)  292300 
Number of pages  9 
Journal  Experimental Mathematics 
Volume  28 
Issue number  3 
Early online date  29 Nov 2017 
DOIs  
Publication status  Published  3 Jul 2019 
Keywords
 algorithms
 finite fields
 functional graphs
 Polynomial maps
 random maps
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Projects


New Applications of Additive Combinatorics in Number Theory and Graph Theory
Mans, B., Shparlinski, I., MQRES, M. & PhD Contribution (ARC), P. C. (.
1/01/14 → 31/12/17
Project: Research