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Abstract
In this paper, we generalize the notion of functional graph. Specifically, given an equation E(X,Y)=0 with variables X and Y over a finite field F_{q} of odd characteristic, we define a digraph by choosing the elements in F_{q} as vertices and drawing an edge from x to y if and only if E(x,y)=0. We call this graph as equational graph. In this paper, we study the equational graph when choosing E(X,Y)=(Y^{2}−f(X))(λY^{2}−f(X)) with f(X) a polynomial over F_{q} and λ a nonsquare element in F_{q}. We show that if f is a permutation polynomial over F_{q}, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials f of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.
Original language  English 

Article number  101667 
Pages (fromto)  131 
Number of pages  31 
Journal  Finite Fields and their Applications 
Volume  64 
DOIs  
Publication status  Published  Jun 2020 
Keywords
 Connected component
 Equational graph
 Finite field
 Functional graph
 Hamiltonian cycle
 Strong connectedness
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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