The number of edges on generalizations of Paley graphs

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THE NUMBER OF EDGES ON GENERALIZATIONS

OF PALEY GRAPHS

LAWRENCE SZE

(Received 21 August 1997)

Abstract.Evans, Pulham, and Sheenan computed the number of complete 4-subgraphs of Paley graphs by counting the number of edges of the subgraph containing only those nodesxfor whichxandx−1 are quadratic residues. Here we obtain formulae for the number of edges of generalizations of these subgraphs using Gaussian hypergeometric series and elliptic curves. Such formulae are simple in several inﬁnite families, including those studied by Evans, Pulham, and Sheenan.

2000 Mathematics Subject Classiﬁcation. 05C30, 11L99.

1. Preliminaries. LetG(p) be a graph with p vertices, and letG(p)c _{denote its} complement, the graph wherexyis an edge if and only ifxyis not an edge ofG(p). Ifp≡1(mod 4)is prime, then letP (p)denote the Paley graph whose vertices are inGF (p), the ﬁnite ﬁeld withpelements. This is the graph wherexyis an edge if and only ifx−yis a quadratic residue modulop. Motivated by a conjecture of Erdös, which turned out to be false, Evans, Pulham, and Sheenan [2] computedk4(P (p))+ k4(P (p)c_{), where}_k

m(G)denotes the number of completem-subgraphs of G. Counting the number of edges ofG∗(p), the subgraph ofP (p)containing only those nodesx for whichxandx−1 are both quadratic residues, was the main obstacle in obtaining their result. They showed [2, Proposition 4] that ifp=4y2₊_x2_{, then the number of} edges inG∗(p)is

p2₋_22p₊_4x2₊₈₁

64 . (1.1)

We compute the number of edges ofG(i, t, p), natural generalizations ofG∗(p). Throughoutpis an odd prime, andGF (p)is the ﬁnite ﬁeld withpelements. Further-more letGF (p)×denote the nonzero elements ofGF (p), and letGF (p)×2denote the nonzero squares. For convenience we letφ(·)denote the Legendre symbol(·/p) ex-tended toGF (p)under the convention thatφ(0):=0. Ifnis an integer, then ordp(n) is the power ofpdividingnand ordp(a/b):=ordp(a)−ordp(b).

Deﬁnition_1.1. _{Let 1}≤_i≤_{8 be an integer. If}_p_{is an odd prime and}_t_{a nonzero}

integer, then the generalized Paley graphG(i, t, p)is the directed graph whose edge setE(i, t, p)is

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The reader should note that edges can be loops, and also that the indexisimply keeps track of the eight nontrivial combinations of signs forφ(1−x),φ(1−y), and φ(x−ty). Ifp≡1 (mod 4), then sinceφ(x−y)=φ(y−x)the graphG(1,1, p)is a double cover ofG∗(p).

We recall the definition of a Gaussian hypergeometric series as defined in [3]. Extend all charactersχofGF (p)×toGF (p)by settingχ(0):=0. IfAandBare two characters ofGF (p), thenA_Bis defined by the normalized Jacobi sum

A B

:=B(−1)

p J(A,B)¯ = B(−1)

p

x∈GF (p)

A(x)B(1¯ −x). (1.3)

Deﬁnition1.2. Given charactersA0, A1, . . . A_n, andB1, B2, . . . B_nofGF (p), let

n+1Fn

_A0, _A1, _{. . . A}

n

B1, . . . Bn|t (1.4)

be the Gaussian hypergeometric series deﬁned by

n+1Fn

_A0, _A1, _{. . . A}

n B1, . . . Bn|

t := p p−1

χ

Aoχ χ

A1χ B1χ

···

Anχ Bnχ

χ(t), (1.5)

where the summation is over all the charactersχofGF (p).

Although the Gaussian hypergeometric series depend on the primep, we suppress its dependence in the notation under the assumption that the prime will be clear from context.

For our purposes it will be important to evaluate

2F1(t):=2F1

_φ _φ

ε|t , 3F2(t):=3F2

_φ _φ _φ

ε ε|t , (1.6)

whereεis the identity (i.e.,ε(x)=1 forx=0). In [3] it was shown that

2F1(t)=φ(_p−1) x∈GF (p)

φ(x)φ(1−x)φ(1−tx), (1.7)

3F2(t)=φ(_p−₂1) x,y∈GF (p)

φ(x)φ(y)φ(1−x)φ(1−y)φ(x−ty). (1.8)

A useful alternative for computing these two Gaussian hypergeometric series was given in [4], where it was shown that they may also be expressed in terms of the number of points on special elliptic curves overGF (p). Deﬁne elliptic curves2E1(t) and3E2(t)by

2E1(t):y2=x(x−1)(x−t),

3E2(t):y2₌_x3₋_t2_x2₊_4t3₋_t4_x₊_t6₋_4t5_. (1.9) Denote the number of points on2E1(t)and3E2(t)overGF (p)by

2N1(t, p):=(x, y)∈GF (p)×GF (p)|y2₌_x(x₋_1)(x₋_t)_,

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Now deﬁne theFrobenius traces,2a1(t, p)and3a2(t, p)by 2a1(t, p):=p−2N1(t, p),

3a2(t, p):=p−3N2(t, p). (1.11)

In this notation, the following two theorems were proved in [4].

Theorem_1.3. _If_t∈Q−{_0,₁}_and_p_{is an odd prime for which}_ord_p_(t(t−₁₎₎=₀_, then

2F1(t)= −φ(−1)2a1(t, p)

p . (1.12)

Theorem1.4. Ifδ∈Q−{0,4}and p is an odd prime for whichord_p(δ(δ−4))=0, then

3F2

₄

4−δ =

φδ2₋_4δ

3a2(δ, p)2−p

p2 . (1.13)

2. Main theorems. Here we compute the number of edges of the graphsG(i, t, p) whent≡0 (modP ). Without loss of generality, we assume that 1≤t≤p−1, although for aesthetic reasons we writet= −1 rather thant=p−1.

Lemma_2.1. _If_p_{is an odd prime and t}≡₀_(mod_p)_{, then}

(i) _{x∈GF (p)}φ(x2₋_t)_{= −}₁_,

(ii) |{x∈GF (p)|φ(x2₋_t)₌₁_{}| =}_(p₋₂₋_φ(t))/2_,

(iii) |{x∈GF (p)|φ(t−x2₎₌₁_{}| =}_(p₋₁₋_φ(t)₋_φ(₋_1))/2_, (iv) |{x∈GF (p)|φ(1−x2_/t)₌₁_{}| =}_(p₋₁₋_φ(t)₋_φ(₋_t))/2_.

Proof. _{(i) By Euler’s criterion that}_φ(x)≡_x(p−1)/2 _(mod_p)_{for all}_x∈_{GF (p),}_and

the Binomial theorem we obtain

x∈GF (p)

φx2−t≡ x∈GF (p)

x2−t(p−1)/2 (modp)

≡

x∈GF (p) (p−1)/2

r=0

(p−1)/2 r

x2r (−1)(p−1)/2−r (modp). (2.1)

Sincex_{x∈GF (p)}k ≡0(modp)for 0< k < p−1, the above sum is

≡

x∈GF (p)

(−t)(p−1)/2+xp−1≡ −1 (modp). (2.2)

Furthermore, this sum is odd because_{x∈GF (p)}φ(x2₋_t)₌_φ(₋_t)₊₂(p−1)/2

x=1 φ(x2−t). Therefore since|x∈GF (p)φ(x2−t)|< p, one easily concludes that the sum is−1.

(ii) DeﬁneS+andS−by

S±:=x∈GF (p)|φx2−t= ±1. (2.3)

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Lemma_2.2. _{Suppose p is an odd prime and}_t≡₀_,₁_(mod_p)_{. If}_γ_±,±_{(t, p)}_{is deﬁned} by

γ±±(t, p):=

x∈GF (p)×|φ1−x2= ±1, φ

1−x_t2 = ±1, (2.4)

then

γ++(t, p)=1₄p+φ(−t)p2F1(t)−φ(−1)−2φ(t)

−φ(−t)−2φt2−t−φ(1−t)−φt−t2−7,

γ+−(t, p)=1₄p−φ(−t)p2F1(t)−φ(−1)−φ(1−t) −φt−t2₊_φ(₋_t)₊_2φ_t2₋_t₋₃_,

γ−+(t, p)=1₄p−φ(−t)p2F1(t)+φ(−1)+φ(1−t) +φt−t2₋_φ(₋_t)₋_2φ_t2₋_t₋₃_,

γ−−(t, p)=1₄p+φ(−t)p2F1(t)+φ(−1)+φ(1−t) +φt−t2+φ(−t)+2φt2−t−2φ(t)−3.

(2.5)

Proof. These formulae follow from four key relations. Sincet≡0,1(modp), it

is clear that

γ++(t, p)+γ−−(t, p)+γ+−(t, p)+γ−+(t, p)

=

x∈GF (p)×|φ1−x2=0, φ

1−x 2 t =0

=p−4−φ(t).

(2.6)

Similarly it is easy to see that

γ++(t, p)+γ−−(t, p)−γ+−(t, p)−γ−+(t, p)

=

x∈GF (p)×

φ1−x2_φ

1−x_t2

= −1+

x∈GF (p)

φ1−x2_φ

1−x2 t .

(2.7)

Sincex2_{represents each nonzero quadratic residue twice, replacing}_x2_by_x_{in the} above expression and then multiplying the summand by the weight (1+φ(x)) leads to

−1+

x∈GF (p)

φ(1−x)φ

1−x

t 1+φ(x)

= −1+

x∈GF (p)

φ(1−x)φ

1−x_t + x∈GF (p)

φ(1−x)φ

1−x_t φ(x).

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This now reduces byLemma 2.1and (1.7) to be

=φ(−t)p2F1(t)−1−φ(t). (2.9)

Now consider

γ++(t, p)+γ+−(t, p)=

x∈GF (p)×|φ1−x2₌_{1, φ}

1−x_t2 =0

=x∈GF (p)×|φ1−x2=1

−x∈GF (p)×|φ1−x2=1, φ

1−x 2 t =0

.

(2.10)

The ﬁrst term is known byLemma 2.1, and the second is 2(1+φ(t)/2)(1+φ(1−t)/2) since it is 2 iftand 1−tare both quadratic residues and 0 otherwise. This leads to

γ₊₊(t, p)+γ₊₋(t, p)=

_p₋₂₋_φ(₋₁₎

2 −1 −2

₁₊_φ(t)

2

1+φ(1−t) 2

=1₂p−φ(−1)−φ(t)−φ(1−t)−φt−t2−5.

(2.11)

Similarly we obtain

γ++(t, p)+γ−+(t, p)=1₂p−φ(t)−φ(−t)−2φt2−t−5. (2.12)

Solving (2.6), (2.9), (2.11), and (2.12) forγ++(t, p),γ+−(t, p),γ−+(t, p), andγ++(t, p) produces the result.

Theorem 2.3depends on auxiliary constants determined by the values ofφ(−1), φ(t), andφ(1−t). These constants are deﬁned in TablesA.1,A.2,A.3,A.4,A.5,A.6,

A.7, andA.8in the appendix.

Theorem2.3. Ifpis an odd prime, and2≤t≤p−1, then

_{E(i, t, p)}

=p2+p2·Ᏺ(i, t, p)·3F2(t)+p·_2F1(t)·Ᏻ(i, t, p)+p·Ᏼ(i, t, p)+᏶(i, t, p)

32 .

(2.13)

Proof. _Deﬁne_{α(i, t, p)}_by

α(i, t, p):=(x, y)∈GF (p)××GF (p)×|φ1−x2₌₍₋₁₎(i−1)/4_, φ1−y2=(−1)(i−1)/2, φx2−ty2=(−1)i−1.

(2.14)

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equations. First,

α(1, t, p)−α(2, t, p)−α(3, t, p)+α(4, t, p)

−α(5, t, p)+α(6, t, p)+α(7, t, p)−α(8, t, p)

=

x,y∈GF (p)×

φ1−x2_φ₁₋_y2_φ_x2₋_ty2

=

x,y∈GF (p)

φ1−x21−y2x2−ty2

−

x,y∈GF (p), xy=0

φ1−x21−y2x2−ty2

=

x,y∈GF (p)

φ(1−x)(1−y)(x−ty)1+φ(x)1+φ(y)

+1+φ(−1)+φ(t)+φ(−t).

(2.15)

The above sum involves the four simpler sums

A:= x,y∈GF (p)

φ(1−x)(1−y)(x−ty),

B:= x,y∈GF (p)

φ(1−x)(1−y)(x−ty)x,

C:= x,y∈GF (p)

φ(1−x)(1−y)(x−ty)y,

D:= x,y∈GF (p)

φ(1−x)(1−y)(x−ty)xy.

(2.16)

ByLemma 2.1,AandBare given by

A= 1/t=y∈GF (p)

φ(1−y) x∈GF (p)

φ(1−x)(x−ty)

+φ

1−1 t

x∈GF (p)

φ(1−x)(x−1)

=φ

₁

t−1 +(p−1)φ

₁

t−1

=φ

₁

t−1 p,

B= 1/t=y∈GF (p)

φy−y2

x∈GF (p)

φ(1−x)(x−ty)

+φ

₁

t− 1 t2

x∈GF (p)

φ(1−x)(x−1)

=

1/t=y∈GF (p)

φy−y2₋_φ(₋₁₎₊_(p₋_1)φ(1₋_t)

=1+φ(1−t)+(p−1)φ(1−t)

=φ(1−t)p+1.

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SimilarlyCcan easily be shown to beC=φ(−t)+φ(1−t)p.By (1.8),Dis a simple multiple of3F2(t), which combined with the formulae forA,B, andCleads to

α(1, t, p)−α(2, t, p)−α(3, t, p)+α(4, t, p)−α(5, t, p)

+α(6, t, p)+α(7, t, p)−α(8, t, p) =φ(−1)3F2(t)p2+φ(1−t)2p

+φt−t2p+2+2φ(−t)+φ(−1)+φ(t).

(2.18)

Eachα(i, t, p)can be expressed in terms ofφ(−1),φ(t),φ(1−t),2F1(t), and3F2(t). Determining these expressions is no more than solving simple systems of equations. For brevity, we only consider the case where(φ(−1), φ(t), φ(1−t))=(1,−1,−1).The solution in this case determines the entries inTable A.5. In the remaining cases, the tables are determined in exactly the same way.

We ﬁrst derive an equation forα(1, t, p)+α(3, t, p)in the following way:

α(1, t, p)+α(3, t, p)

=(x, y)∈GF (p)××GF (p)×|φ1−x2₌_{1, φ}₁₋_y2₌_{0, φ}_x2₋_ty2₌₁ =(x, y)∈GF (p)××GF (p)×|φ1−x2=1, φx2−ty2=1

−2

x∈GF (p)|φ1−x2=1, φ

1−x 2

t =φ(−t)= −1

=x∈GF (p)×|φ1−x2₌₁_·_y_∈_{GF (p)}×_|_φ₁₋_ty2₌₁₋_2γ +−(t, p)

=

_p₋₅

2

p−1

2 −

p−7+p2F1(t)

2 .

(2.19)

The following equations are determined in a similar way,

α(1, t, p)+α(2, t, p)=

_p₋₅

2 2

,

α(3, t, p)+α(4, t, p)=

_p₋₅

2

p−1 2 ,

α(5, t, p)+α(6, t, p)=

_p₋₁

2

p−5 2 ,

α(7, t, p)+α(8, t, p)=

_p₋₁

2 2

−p+1−p2F1(t)

2 ,

α(1, t, p)+α(5, t, p)=

_p₋₅

2

p−1

2 −

p−3+p2F1(t)

2 ,

α(3, t, p)+α(7, t, p)=

_p₋₁

2 2

−p−1−p2F1(t)

2 .

(2.20)

The solution to the system (2.18), (2.19), and (2.20) forα(i, t, p)for 1≤i≤8 yields the entries inTable A.5.

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Corollary_2.4. _If_p_{is an odd prime,}₂≤_t≤_p−₁_{, and}_δ_:≡_(4t−_{4)/t (mod}_p)_{, then}

_{E(i, t, p)}₌p2 32

φ(1−t)Ᏺ(i, t, p)+1

+ p 32

−φ(−1)Ᏻ(i, t, p)+Ᏼ(i, t, p)

−2φ(1−t)Ᏺ(i, t, p)3N2(δ, p)−φ(1−t)Ᏺ(i, t, p)

+₃₂1 φ(1−t)Ᏺ(i, t, p)3N2(δ, p)2

+φ(−1)Ᏻ(i, t, p)2N1(t, p)+᏶(i, t, p).

(2.21)

Example_2.5. _{Consider the graph}_G(7,4,13)._By_{Corollary 2.4}_{we ﬁnd that}_δ≡₃

(mod 13), and by (1.9) we are lead to consider the GF(13) points of the curves

2E1(4):y2=x3−5x2+4x, 3E2(3):y2=x3−9x2+27x−243. (2.22)

Both curves have2N1(4,13)= 3N2(3,13)=15 points over GF(13), and so byCorollary 2.4(usingTable A.1sinceφ(−1)=φ(4)=φ(−3)=1) we ﬁnd that|E(7,4,13)| =4. It is easy to verify that this is indeed true, since the edge set is

E(7,4,13)= {3 →3,9 →3,9→9,12 →12}. (2.23)

Ift≡ −1 (modp), then formulae like (1.1) follow from evaluations proved in [4]

2F1(−1)=

      

0, ifp≡3(mod 4); 2x(−1)(x+y+1)/2

p , ifp≡1(mod 4), x

2₊_y2₌_{p, x}_odd.

3F2(−1)=

          

−φ(2)

p , ifp≡5,7(mod 8); φ(2)4x2₋_p

p2 , ifp≡1,3(mod 8), x

2₊_2y2₌_p.

(2.24)

Corollary_2.6. _Let_p_{be prime, and deﬁne integers}_u_,_v_,_x_{, and}_y_by

p=x2₊_y2_, _if_p_≡₁_{(mod 4)}_with_x_odd_,

p=u2+2v2, ifp≡1,3(mod 8). (2.25)

Ifp≡1(mod 8), then

E(i,−1, p)= 1 32·

                

p2₋_12p₋_12x(₋₁₎(x+y+1)/2₊_4u2₊_91, _if_i₌_1; p2₋_12p₊_4x(₋₁₎(x+y+1)/2₋_4u2₊_19, _if_i₌_1,3_or_5; p2₋_4p₊_4x(₋₁₎(x+y+1)/2₊_4u2₊_3, _if_i₌_4,6_or_7; p2₋_4p₋_12x(₋₁₎(x+y+1)/2₋_4u2₋_5, _if_i₌_8.

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E(i,−1, p)= 1 32·               

p2₋_10p₊_4u2₊_17, _if_i₌₁_or_7; p2₋_2p₋_4u2₊_1, _if_i₌₂_or_8; p2₋_6p₋_4u2₊_13, _if_i₌₃_or_5; p2₋_6p₊_4u2₊_5, _if_i₌₄_or_6.

(2.27)

E(i,−1, p)= 1 32·         

p2₋_16p₋_12x(₋₁₎(x+y+1)/2₊_67, _if_i₌_1; p2₋_8p₊_4x(₋₁₎(x+y+1)/2₊_11, _if₂_≤_i_≤_7; p2₋_12x(₋₁₎(x+y+1)/2₊_19, _if_i₌_8.

(2.28)

E(i,−1, p)= 1 32·               

p2₋_6p₊_25, _if_i₌₁_or_7; p2₋_6p₋_7, _if_i₌₂_or_8; p2₋_10p₊_21, _if_i₌₃_or_5; p2₋_2p₋_3, _if_i₌₄_or_6.

(2.29)

Even though the onlytfor which2F1(t)and 3F2(t)are known to simultaneously have explicit evaluations aret=0 and±1, we can still obtain simple formulae using the fact thatᏳ(i, t, p)is often zero in TablesA.3,A.5,A.7, andA.8. Ifp >3 is prime, then the following formulae were obtained in [4]:

3F2(−8)=

         −1

p, ifp≡3(mod 4); 4x2₋_p

p2 , ifp≡1(mod 4), x

2₊_y2₌_{p, x}_odd.

3F2 ₋₁ 8 =         

−φ(2)_p , ifp≡3(mod 4);

φ(2)4x2₋_P

p2 , ifp≡1(mod 4), x

2₊_y2₌_{p, x}_odd.

(2.30)

Now as an immediate consequence we obtain the following result.

Corollary2.7. Suppose thatp≡5or7 (mod 8)is prime, and thatt≡ −8or−1/8

(modp). Ifp≡5 (mod 8)andt≡ −8(modp), then

_{E(i, t, p)}₌ 1 32·

  

p2₋_6p₋_4x2₊_9, _if_i₌₃_or_5;

p2₋_6p₊_4x2₊_1, _if_i₌₄_or_6. (2.31)

Ifp≡5(mod 8)andt≡ −1/8(modp), then

E(i, t, p)=₃₂1 ·

  

p2₋_6p₊_4x2₊_9, _if_i₌₃_or_5;

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Ifp≡7(mod 8)andt≡ −8or−1/8, then

E(i, t, p)= 1 32·

  

p2₋_6p₊_25, _if_i₌₁_or_7;

p2₋_6p₋_7, _if_i₌₂_or_8. (2.33)

Example 2.8. Consider the graphG(8,−1,17). Since 17= 12+42=32+2·22, Corollary 2.6 withx =1, y=4, u=3, and v=2 also shows|E(8,−1,17)| =6. It is easy to check that the 6 edges are

E(8,−1,17)= {4→8,8 →4,15→13,15 →8,13→15,8 →15}. (2.34)

Example2.9. Consider the graphG(2,15,23). Since 23≡7(mod 8)and 15≡ −8

(mod 23),Corollary 2.7implies that|E(2,−8,23)| =12. It is easy to check that these 12 edges are

E(2,15,23)= {16→8, 16→12,18→6,18 →12,16→18,12→6,

12→8, 8→6,8 →18,8→16,6 →12,6→16}. (2.35)

Now we state the result whent=1. Since the proof in this case follows the same type of argument leading toTheorem 2.3, we omit it for brevity.

Theorem_2.10. _{If p is an odd prime, then the number of edges of}_{G(i,1, p)}_is

_{E(i,1, p)}₌ 1 32·

                    

p2₊_p2_φ(₋₁₎_3F2(1)₋_15p₋_φ(₋₁_)5p₊_φ(₋₁₎₃₀₊_51, _if_i₌_1; p2₋_p2_φ(₋₁₎_3F2(1)₋_9p₊_φ(₋_1)p₋_φ(₋₁₎₆₊_15, _if_i₌_2,₅_or_8; p2₋_p2_φ(₋₁₎_3F2(1)₋_5p₋_φ(₋_1)3p₊_φ(₋₁₎₆₊_3, _if_i₌_3; p2₊_p2_φ(₋₁₎_3F2(1)₋_7p₊_φ(₋_1)3p₋_φ(₋₁₎₁₀₊_11, _if_i₌₄_or_7; p2₊_p2_φ(₋₁₎_3F2(1)₋_3p₋_φ(₋_1)p₊_φ(₋₁₎₂₋_1, _if_i₌_6.

(2.36)

Since Evans (see [1]) proved that3F2(1)=(4x2₋_2p)p2_if_p₌_x2₊_y2_where_x _is odd, and is zero otherwise, we obtain the following corollary.

Corollary2.11. Ifp=x2+y2is prime wherexis odd, then

_{E(i,1, p)}₌ 1 32·

        

p2₋_22p₊_4x2₊_81, _if_i₌_1;

p2₋_6p₋_4x2₊_9, _if_i₌_2,3,5,_or_8; p2₋_6p₊_4x2₊_1, _if_i₌_4,6,_or_7.

(2.37)

If p≡3(mod 4)is prime, then

E(i,1, p)=₃₂1 ·

  

p2₋_10p₊_21, _if_i₌_1,2,4,5,7,_or_8;

p2₋_2p₋_3, _if_i₌₃_or_6. (2.38)

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Appendix

TableA.1. φ(−1), φ(t), φ(1−t)=(1,1,1).

i Ᏺ(i, t, p) Ᏻ(i, t, p) Ᏼ(i, t, p) ᏶(i, t, p)

1 1 −6 −11 91

2 −1 2 −13 19

3 −1 2 −13 19

4 1 2 −3 3

5 −1 2 −13 19

6 1 2 −3 3

7 1 2 −3 3

8 −1 −6 −5 −5

TableA.2. φ(−1), φ(t), φ(1−t)=(−1,1,1).

1 −1 2 −5 27

2 1 2 −11 19

3 1 −6 −11 19

4 −1 2 −5 27

5 1 2 −11 19

6 −1 −6 −5 −5

7 −1 2 −5 27

8 1 2 −11 19

TableA.3. φ(−1), φ(t), φ(1−t)=(1,−1,1).

1 1 −4 −9 29

2 −1 4 −11 21

3 −1 0 −7 9

4 1 0 −5 1

5 −1 0 −7 9

6 1 0 −5 1

7 1 4 −1 5

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TableA.4. φ(−1), φ(t), φ(1−t)=(1,1,−1).

1 1 −6 −17 67

2 −1 2 −7 11

3 −1 2 −7 11

4 1 2 −9 11

5 −1 2 −7 11

6 1 2 −9 11

7 1 2 −9 11

8 −1 −6 1 19

TableA.5. φ(−1), φ(t), φ(1−t)=(1,−1,−1).

1 1 −4 −11 21

2 −1 4 −9 29

3 −1 0 −5 1

4 1 0 −7 9

5 −1 0 −5 1

6 1 0 −7 9

7 1 4 −3 −3

8 −1 −4 −1 5

TableA.6. φ(−1), φ(t), φ(1−t)=(−1,1,−1).

1 −1 2 −11 19

2 1 2 −5 27

3 1 −6 −5 −5

4 −1 2 −11 19

5 1 2 −5 27

6 −1 −6 −11 19

7 −1 2 −11 19

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TableA.7. φ(−1), φ(t), φ(1−t)=(−1,−1,1).

1 −1 0 −7 25

2 1 0 −5 −7

3 1 −4 −9 21

4 −1 4 −3 −3

5 1 4 −9 21

6 −1 −4 −3 −3

7 −1 0 −7 25

8 1 0 −5 −7

TableA.8. φ(−1), φ(t), φ(1−t)=(−1,−1,−1).

1 −1 0 −9 17

2 1 0 −3 1

3 1 −4 −7 13

4 −1 4 −5 5

5 1 4 −7 13

6 −1 −4 −5 5

7 −1 0 −9 17

8 1 0 −3 1

Acknowledgement. _{The author wishes to express thanks to Ken Ono for}

sug-gesting this investigation and for numerous helpful suggestions during the prepara-tion of this article.

References

[1] R. J. Evans,Identities for products of Gauss sums over ﬁnite ﬁelds, Enseign. Math. (2)27

(1981), no. 3-4, 197–209 (1982).MR 83i:10050. Zbl 491.12020.

[2] R. J. Evans, J. R. Pulham, and J. Sheehan,On the number of complete subgraphs contained in certain graphs, J. Combin. Theory Ser. B30(1981), no. 3, 364–371.MR 83c:05075. Zbl 475.05049.

[3] J. Greene,Hypergeometric functions over ﬁnite ﬁelds, Trans. Amer. Math. Soc.301(1987), no. 1, 77–101.MR 88e:11122. Zbl 629.12017.

[4] K. Ono,Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc.350(1998), no. 3, 1205–1223.MR 98e:11141. Zbl 910.11054.

Lawrence Sze: Department of Mathematics, California Polytechnic University,

San Luis Obispo, CA93407, USA