Parameters of Quadratic Residue Digraphs over Certain Finite Fields


  • Louis Beaugris School of Mathematical Sciences, Kean University Union, New Jersey 07083



Quadratic Residues, Digraphs, Trees, Acyclic digraphs, Diameter, Eccentricity of a vertex


Linking graph theory and algebra has been a rich area of mathematical exploration for a long time. Cayley digraphs and Zero-Divisor graphs are two such examples. In this paper, we make another connection by constructing and studying digraphs whose vertices are the elements of the multiplicative group of the finite fields \(\mathbb{Z}_{p}\) for certain primes \(p\). In particular, we determine parameters, including the diameter of such digraphs and the eccentricity of certain vertices of these digraphs. We also find some results on the quadratic residues and nonresidues of \(\mathbb{Z}_{p}\).


Download data is not yet available.


C. Aebi and G. Cairns, Sums of quadratic residues and nonresidues, The American Mathematical Monthly 124(2) (2017), 166 – 169, DOI: 10.4169/amer.math.monthly.124.2.166.

D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, Journal of Algebra 217 (1999), 434 – 447, DOI: 10.1006/jabr.1998.7840.

D.K. Basnet and J. Bhattacharyya, Nil clean graphs of rings, Algebra Colloquium 24(3) (2017), 481 – 492, DOI: 10.1142/S1005386717000311.

I. Beck, Coloring of commutative rings, Journal of Algebra 116 (1988), 208 – 226, DOI: 10.1016/0021-8693(88)90202-5.

A. Cayley, Desiderata and suggestions: no. 2. The theory of groups: Graphical representation, American Journal of Mathematics 1(2) (1878), 174 – 176, DOI: 10.2307/2369306.

G. Chartrand, L. Lesniak and P. Zhang, Graphs and Digraphs, 6th edition, Chapman and Hall/CRC (2015).

A. Das, On nonzero component graph of vector spaces over finite fields, Journal of Algebra and Its Applications 16(1) (2017), 1 – 10, DOI: 10.1142/S0219498817500074.

R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, revised edition, Cambridge University Press (1994).

K. Rosen, Elementary Number Theory and its Applications, 5th edition, Pearson (2010).




How to Cite

Beaugris, L. (2019). Parameters of Quadratic Residue Digraphs over Certain Finite Fields. Communications in Mathematics and Applications, 10(1), 31–36.



Research Article