Constructions of Strongly Regular Cayley Graphs Using Even Index Gauss Sums

Abstract

In this paper, generalizing the result in [9], I construct strongly regular Cayley graphs by using union of cyclotomic classes of Fq and Gauss sums of index w, where w≥2 is even. In particular, we obtain three infinite families of strongly regular graphs with new parameters.

Strongly regular Cayley graphs, skew Hadamard difference sets, and rationality of relative Gauss sums

“…Although it seems that the assumptions of our main result are simpler and the situation is definitely much more general than that of [27], the approach in [27] is obviously different from ours and his results are not completely included in ours. In fact, Wu [27] obtained two conditions (one is an equation and the other is a congruence) which are necessary and sufficient for the construction to give rise to an srg, and his approach has the advantage of revealing an interesting connection between strongly regular Cayley graphs Cay(F p (p 1 −1)/e , C (p (p 1 −1)/e ,p1) 0…”

“…On the other hand, Wu [27] gave necessary and sufficient conditions for Γ to be an srg by generalizing the method used in the paper of Ge, Xiang, and Yuan [13]. Although it seems that the assumptions of our main result are simpler and the situation is definitely much more general than that of [27], the approach in [27] is obviously different from ours and his results are not completely included in ours. In fact, Wu [27] obtained two conditions (one is an equation and the other is a congruence) which are necessary and sufficient for the construction to give rise to an srg, and his approach has the advantage of revealing an interesting connection between strongly regular Cayley graphs Cay(F p (p 1 −1)/e , C (p (p 1 −1)/e ,p1) 0 ) and cyclic difference sets in (Z/p 1 Z, +), which will be very effective to get some new cyclic difference sets and also a strong necessary condition for the existence of cyclotomic srgs.…”

“…On the other hand, Wu [27] gave necessary and sufficient conditions for Γ to be an srg by generalizing the method used in the paper of Ge, Xiang, and Yuan [13]. Although it seems that the assumptions of our main result are simpler and the situation is definitely much more general than that of [27], the approach in [27] is obviously different from ours and his results are not completely included in ours.…”

“…We close this paper by referring the reader to the interesting paper [27] by Wu. Immediately after writing up this manuscript, the author became aware that Wu [27] obtained a nice result on the existence problem of cyclotomic srgs.…”

Quadratic Form Gauss Sums

“…One sees their use in coding theory [37], [116]; point counting on elliptic curves [97], [117]; determining the number of solutions to polynomial equations [7], [95]; determining the value of cyclotomic numbers [114]; constructing difference sets [39]; and the evaluation of various related functions [85]. In particular, the finite field Gauss sums are widely used in the construction of Cayley graphs [38], [40], [43], [87], [88], [113], [115].…”