Projects per year

### 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 non-square 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 |
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Article number | 101667 |

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

## Fingerprint Dive into the research topics of 'On the equational graphs over finite fields'. Together they form a unique fingerprint.

## 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

## Cite this

*Finite Fields and their Applications*,

*64*, [101667]. https://doi.org/10.1016/j.ffa.2020.101667