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Discrete Mathematics
journal homepage:www.elsevier.com/locate/disc
On symplectic graphs modulo p
n
Yotsanan Meemark
∗, Thanakorn Prinyasart
Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand
a r t i c l e
i n f o
Article history:
Received 26 September 2010 Received in revised form 20 April 2011 Accepted 5 May 2011
Keywords:
Graph automorphisms Stable range one Symplectic graphs
a b s t r a c t
In this work, we study a family of regular graphs using the 2ν×2νsymplectic group modulo
pn, where p is a prime and n andνare positive integers. We find that this graph is strongly regular only whenν = 1. In addition, we define the symplectic graphs of a symplectic space V over a commutative ring R and show that it is vertex transitive and edge transitive when R has stable range one, which is the case forZpn.
© 2011 Elsevier B.V. All rights reserved.
1. Symplectic graphs modulo pn
Let p be a prime, and let n be a positive integer. For
ν ≥
1, let V(2ν) denote the set of 2ν
-tuples(
a1,
a2, . . . ,
a2ν)
ofelements inZpn such that aiis invertible modulo pnfor some i
∈ {
1,
2, . . . ,
2ν}
. Define an equivalence relation∼
pn onV(2ν)by
(
a1,
a2, . . . ,
a2ν) ∼
pn(
b1,
b2, . . . ,
b2ν) ⇔ (
a1,
a2, . . . ,
a2ν) = λ(
b1,
b2, . . . ,
b2ν)
for some
λ ∈
Z×pn. HereZ ×pn stands for the unit group modulo pn. Write
[
a1,
a2, . . . ,
a2ν]
for the equivalence class of(
a1,
a2, . . . ,
a2ν)
modulo∼
pn, and let V∼(2ν)pn be the set of all such equivalence classes. Let K
(2ν)be the 2
ν ×
2ν
nonsingularalternate matrix overZ/pnZgiven by K(2ν)
=
[
0 Iν−
Iν 0]
2ν×2ν,
where Iνis the
ν × ν
identity matrix.The symplectic graph modulo pnon V∼(2pnν)relative to K(
2ν), denoted by Sp(2ν)
pn , is the graph whose vertex set is V(
2ν) ∼pn and with adjacency defined by
[
a1,
a2, . . . ,
a2ν]
is adjacent to[
b1,
b2, . . . ,
b2ν]
⇔
(
a1,
a2, . . . ,
a2ν)
K(2ν)(
b1,
b2, . . . ,
b2ν)
t∈
Z × pn⇔
(
a1bν+1−
aν+1b1) + (
a2bν+2−
aν+2b2) + · · · + (
aνb2ν−
a2νbν)
is invertible modulo pn.
The above adjacency condition is well defined. For, if
(
a1,
a2, . . . ,
a2ν) = λ(
a′1,
a2′, . . . ,
a′2ν)
and(
b1,
b2, . . . ,
b2ν) =
µ(
b′ 1,
b ′ 2, . . . ,
b ′ 2ν)
for someλ, µ ∈
Z × pn, thenλµ ∈
Z × pnand(
a1bν+1−
aν+1b1) + (
a2bν+2−
aν+2b2) + · · · + (
aνb2ν−
a2νbν)
=
λµ[(
a′1b′ν+1−
aν+′ 1b′1) + (
a′2b′ν+2−
a′ν+2b′2) + · · · + (
a′νb′2ν−
a′2νb′ν)].
∗ Corresponding author.E-mail addresses:yotsanan.m@chula.ac.th(Y. Meemark),thanakorn_dpst@hotmail.com(T. Prinyasart). 0012-365X/$ – see front matter©2011 Elsevier B.V. All rights reserved.
A simple counting shows that
|
V(2ν)| =
(
pn)
2ν−
(
pn−1)
2νand|
V(2ν)∼pn
| =
(pn)2ν−(pn−1)2ν
pn−pn−1 . Furthermore, we have the following results.
Theorem 1.1. Let p be a prime, and let n and
ν
be positive integers. The symplectic graph Sp(p2nν)is(
pn)
2ν−1-regular and everytwo adjacent vertices of Sp(p2nν)has
(
pn)
2ν−2(
pn−
pn−1)
common neighbors.Proof. Assume that
[
a1,
a2, . . . ,
a2ν]
and[
b1,
b2, . . . ,
b2ν]
are adjacent. Since(
a1,
a2, . . . ,
a2ν) ∈
V(2ν), there exists an i∈ {
1,
2, . . . ,
2ν}
such that aiis invertible modulo pn. If i≤
ν
, thenbν+i
=
a−1
i
(
r+
(
aν+1b1−
a1bν+1) + (
aν+2b2−
a2bν+2) + · · · + (
aν+i−1bi−1−
ai−1bν+i−1)
+
aν+ibi+
(
aν+i+1bi+1−
ai+1bν+i+1) + · · · + (
a2νbν−
aνb2ν))
for some r
∈
Z×pnand if i≥
ν +
1, thenbi−ν
=
a −1i
((
a1bν+1−
aν+1b1) + (
a2bν+2−
aν+2b2) + · · · + (
ai−1−νbi−1−
ai−1bi−1−ν)
+
ai−νbi+
(
ai+1−νbi+1−
ai+1bi+1−ν) + · · · + (
aνb2ν−
a2νbν) −
s)
for some s
∈
Z×pn. Therefore, there are(pn)2ν−1(pn−pn−1) pn−pn−1
=
(
pn
)
2ν−1classes adjacent to the vertex[
a1
,
a2, . . . ,
a2ν]
, and henceSp(p2nν)is
(
pn)
2ν−1-regular.Next, let
[
x1,
x2, . . . ,
x2ν]
be a common neighbor of[
a1,
a2, . . . ,
a2ν]
and[
b1,
b2, . . . ,
b2ν]
. Then(
a1xν+1−
aν+1x1) + (
a2xν+2−
aν+2x2) + · · · + (
aνx2ν−
a2νxν) =
r ′ (1.1) and(
b1xν+1−
bν+1x1) + (
b2xν+2−
bν+2x2) + · · · + (
bνx2ν−
b2νxν) =
s′ (1.2) for some r′,
s′∈
Z×pn. Since p-aiand we may assume without loss of generality that i
≤
ν
, from Eq.(1.1)we havexν+i
=
a −1 i(
r ′+
(
aν+1x1−
a1xν+1) + (
aν+2x2−
a2xν+2) + · · · + (
aν+i−1xi−1−
ai−1xν+i−1)
+
aν+ixi+
(
aν+i+1xi+1−
ai+1xν+i+1) + · · · + (
a2νxν−
aνx2ν)).
Substituting the value of xν+iinto ai
×
(1.2)and subtracting from bi×
(1.1)give−
ν−
j=1(
aibν+j−
aν+ibi)
xj+
ν−
j=1(
aibj−
ajbi)
xν+j=
ais′−
bir′.
Suppose that p
|
(
aibj−
ajbi)
for all j∈ {
1,
2, . . . ,
2ν}
. Then we can prove that p|
(
akbl−
albk)
for all k,
l∈ {
1,
2, . . . ,
2ν}
. This implies p| [
(
a1b2−
a2b1) + (
a3b4−
a4b3) + · · · + (
a2ν−1b2ν−
a2νb2ν−1)]
which contradicts[
a1,
a2, . . . ,
a2ν]
is adjacentto
[
b1,
b2, . . . ,
b2ν]
. Thus, there exists a j∈ {
1,
2, . . . ,
2ν}
such that(
aibj−
ajbi) ∈
Z×
pn. Hence, there are
(
pn)
2ν−2(
pn−
pn−1)(
pn−
pn−1)
pn−
pn−1=
(
pn
)
2ν−2(
pn−
pn−1)
classes of common neighbors of adjacent vertices
[
a1,
a2, . . . ,
a2ν]
and[
b1,
b2, . . . ,
b2ν]
.Corollary 1.2. The symplectic graph Sp(p2)is a complete graph.
A strongly regular graph with parameters
(v,
k, λ, µ)
is a k-regular graph onv
vertices such that for every pair of adjacent vertices there areλ
vertices adjacent to both, and for every pair of non-adjacent vertices there areµ
vertices adjacent to both. Therefore,Theorem 1.1shows that our symplectic graphs satisfy three parametersv =
(pn)2ν−(pn−1)2νpn−pn−1
,
k=
(
p n)
2ν−1and
λ = (
pn)
2ν−2(
pn−
pn−1)
. Unfortunately, the last parameterµ
is fulfilled only whenν =
1. Recall from Chapter 10 of [1] that the eigenvalues of a non-complete connected strongly regular graph with parameters(v,
k, λ, µ)
are (λ−µ)±√ ∆
2 and k
and the corresponding multiplicities are12
v −
1±
(v−1)(µ−λ)−√ 2k ∆
and 1, where
∆ =
(λ − µ)
2+
4(
k−
µ) >
0.Theorem 1.3. For n
≥
2, the symplectic graph Sp(p2n) is strongly regular with parameters(
pn+
pn−1,
pn,
pn−
pn−1,
pn)
.Consequently, its eigenvalues are
−
pn−1,
0 and pnwith multiplicities pn+
pn−1−
p−
1,
p and 1, respectively.To prove this theorem, it remains to verify the final parameter
µ
. This will be a consequence of the following two lemmas. The first one talks about the representatives for the vertex set V∼(2pn), which can be partitioned into p+
1 sets of the sameLemma 1.4. Let p be a prime and n a positive integer. For k
∈ {
0,
1, . . . ,
p−
1}
, let Ak= {[
1,
pi+
k] :
i∈ {
0,
1,
2, . . . ,
pn−1
−
1}}
,
and let Ap
= {[
j,
1] :
0≤
j<
pnand p|
j}
.
Then
|
Ak| =
pn−1for all k∈ {
0,
1, . . . ,
p}
and the collection{
Ak:
k∈ {
0,
1, . . . ,
p}}
is a partition of V∼(2pn), that is, Ak∩
Al= ∅
for all k
,
l∈ {
0,
1, . . . ,
p}
and k̸=
l, and V(2)∼pn
=
p k=0Ak.The second lemma gives a necessary and sufficient condition for adjacent vertices of Sp(p2n), and a conclusion for
Theorem 1.3. Observe that for k
∈ {
0,
1, . . . ,
p−
1}
, [
1,
pi+
k]
is not adjacent to[
1,
pj+
k]
for all i,
j∈ {
0,
1, . . . ,
pn−2}
and[
i,
1]
is not adjacent to[
j,
1]
for all 0≤
i,
j<
pn,
p|
i and p|
j. Moreover, we prove the following lemma.Lemma 1.5. Let p be a prime and n a positive integer. Let k
,
l∈ {
0,
1, . . . ,
p}
, [
a1,
a2] ∈
Akand[
b1,
b2] ∈
Al. Then k̸=
l if andonly if
[
a1,
a2]
is adjacent to[
b1,
b2]
. In addition, every two non-adjacent vertices of Sp( 2)pn have pncommon neighbors.
Proof. The above observation shows the necessity. To prove the sufficiency, we may assume without loss of generality that
k
<
l. If l=
p, then[
a1,
a2] = [
1,
pi+
k]
for some i∈ {
0,
1,
2, . . . ,
pn−1−
1}
and[
b1,
b2] = [
j,
1]
for some 0≤
j<
pnand p
|
j, so[
a1,
a2]
is not adjacent to[
b1,
b2]
caused by p -(
1−
(
pi+
k)
j)
. Next, we suppose that k<
l<
p. Then[
a1,
a2] = [
1,
pi+
k]
and[
b1,
b2] = [
1,
pj+
l]
for some i,
j∈ {
0,
1,
2, . . . ,
pn−1−
1}
. Since 0≤
k<
l<
p,
p does not divide p(
j−
i) +
l−
k, and hence[
a1,
a2]
is not adjacent to[
b1,
b2]
as desired.Finally, we show the final statement which completes the proof ofTheorem 1.3. Suppose that
[
a1,
a2]
and[
b1,
b2]
arenon-adjacent vertices. Then both of them belong to Alfor some l
∈ {
0,
1, . . . ,
p}
. Thus, the set of their common neighbors is V∼(2pn) rAlwhich consists of pnelements.Lemma 1.5also gives an immediate corollary.
Corollary 1.6. Let p be a prime and n a positive integer. The symplectic graph Sp(p2n)is a
(
p+
1)
-partite graph with the partition{
Ak:
k∈ {
0,
1, . . . ,
p}}
. As a result, its chromatic number is p+
1.Since Sp(p2n) is a complete
(
p+
1)
-partite regular graph whose vertex set is partitioned by the collection{
Ak:
k∈
{
0,
1, . . . ,
p}}
, an automorphism of this graph corresponds with a permutation of{
A0,
A1, . . . ,
Ap}
, and a permutation of vertices in Akfor all k=
0,
1, . . . ,
p. Hence, we have the following theorem.Theorem 1.7. Let p be a prime and n a positive integer. Then
|
Aut Sp(p2n)| =
(
pn−1
!
)
p+1(
p+
1)!
.We conclude this section with an example showing that if
ν ≥
2, then the parameterµ
for the symplectic graph Sp(p2nν)is not fulfilled, and thus it is not strongly regular.
Example 1.8. Let
ν ≥
2. Consider three vertices⃗
a= [
1,
0,
0, . . . ,
0]
,b⃗
= [
1,
0, . . . ,
0,
p,
0, . . . ,
0]
, where p is at theν +
1st entry, and⃗
c= [
1,
1,
0,
0, . . . ,
0]
in V∼(2pnν). It is easy to see that⃗
a is not adjacent to⃗
b and⃗
b is not adjacent to⃗
c. Assume that[
x1,
x2, . . . ,
x2ν]
is a common neighbor for⃗
a and⃗
b. Then p must divide xν+1and other xi’s are free. This implies that there are(
pn−
pn−1)
pn(2ν−1)distinct classes of common neighbors for⃗
a and⃗
b. On the other hand, suppose that[
y1
,
y2, . . . ,
y2ν]
is acommon neighbor for
⃗
b and⃗
c. Thus we must have gcd(
yν+1,
p) =
gcd(
yν+1+
yν+2,
p) =
1 and other yj’s are free. This yields(
pn−
pn−1)
2pn(2ν−2)distinct classes of common neighbors forb and⃗
⃗
c. Hence, there are two pairs of non-adjacent verticessuch that the numbers of common neighbors of them are not the same, and so the symplectic graph Sp(p2nν)is not strongly
regular.
Remark. Tang and Wan [3] worked on the general symplectic graph Sp
(
2ν,
q)
over the finite fieldFq. This graph is strongly regular with parameters
q2qν−−11,
q2ν−1,
q2ν−2(
q−
1),
q2ν−2(
q−
1)
. Their proof used orthogonal complements and matrixtheory over finite fields. In contrast, our symplectic graph is defined analogously over the commutative ringZpn and our
proof uses combinatorial method.
2. Vertex transitive and edge transitive
Let R be a commutative ring and let V be a free R-module of R-dimension n, where n
≥
2. Assume that we have a functionβ :
V×
V→
R which is R-bilinear,β(⃗
x, ⃗
x) =
0 for all⃗
x∈
V and the R-module morphism from V to V∗=
HomR(
V,
R)
given by⃗
x→
β(·, ⃗
x)
is an isomorphism. We call the pair(
V, β)
a symplectic space.A vector
⃗
x in V is said to be unimodular if there is an f in V∗with f(⃗
x) =
1; equivalently, if⃗
x=
α
1
⃗
b1+ · · · +
α
nb⃗
nwhere{⃗
b1, . . . , ⃗
bn}
is a basis for V andα
1, . . . , α
n∈
R, then the ideal(α
1, . . . , α
n) =
R. If⃗
x is unimodular, then the line Rx is a freeLemma 2.1. Let
⃗
x and⃗
y be unimodular vectors in V . Then R⃗
x=
Ry if and only if⃗
⃗
x=
λ⃗
y for someλ ∈
R×.Proof. Assume that R
⃗
x=
R⃗
y. Then⃗
x=
λ⃗
y for someλ ∈
R. Since⃗
x is unimodular, there is an f∈
V∗such that f(⃗
x) =
1. Thus, 1=
f(⃗
x) =
f(λ⃗
y) = λ
f(⃗
y)
, soλ
is a unit in R. The converse is clear.A hyperbolic pair
{⃗
x, ⃗
y}
is a pair of unimodular vectors in V with the property thatβ(⃗
x, ⃗
y) =
1. The module R⃗
x⊕
R⃗
y is called a hyperbolic plane. Let(
V, β)
be a symplectic space. An R-module automorphism T on V is an isometry on V ifβ(
T(⃗
x),
T(⃗
y)) = β(⃗
x, ⃗
y)
for all⃗
x, ⃗
y∈
V . The group of isometries on V is called the symplectic group of V over R and denoted by SpR(
V)
.Define the graphGSpR(V)with the vertex set is the set of lines
{
R⃗
x: ⃗
x is a unimodular in V}
and with adjacency given byR
⃗
x is adjacent to R⃗
y if and only ifβ(⃗
x, ⃗
y) ∈
R×(equivalently,β(⃗
x, ⃗
y) =
1). We callGSpR(V), the symplectic graph of V over R. To see that this adjacency condition is well defined, let⃗
x1, ⃗
x2, ⃗
y1and⃗
y2be unimodular vectors in V . Assume that R⃗
x1=
R⃗
x2and Ry
⃗
1=
R⃗
y2. ByLemma 2.1,⃗
x1=
λ⃗
x2and⃗
y1=
λ
′y⃗
2for someλ, λ
′∈
R×, and hence we haveβ(⃗
x1, ⃗
y1) ∈
R ×⇔
β(λ⃗
x2, λ
′⃗
y2) ∈
R ×⇔
λλ
′β(⃗
x2, ⃗
y2) ∈
R ×⇔
β(⃗
x2, ⃗
y2) ∈
R ×.
A commutative ring R is said to have stable range one if for all
α, β ∈
R with(α, β) =
R, there exists aδ
in R such thatα + δβ ∈
R×. Kirkwood and McDonald (Theorem 2.8 of [2]) showed the following results.Lemma 2.2 ([2]). Let R be a commutative ring with stable range one and 2 a unit. Let V be a symplectic space over R. Then,
SpR
(
V)
acts transitively on unimodular vectors and on hyperbolic planes.We shall apply the above lemma to verify that our symplectic graph is vertex transitive and edge transitive (that is, its automorphism group acts transitively on the vertex set and the edge set, respectively) when R has stable range one and 2 is a unit.
Theorem 2.3. Let R be a commutative ring with stable range one and 2 a unit. Then the symplectic graphGSpR(V) is vertex
transitive and edge transitive.
Proof. Observe that for any automorphism T of V , we have the induced automorphism
σ
Ton the vertex set of the symplecticgraphGSp(V)given by
σ
T:
R⃗
x→
RT(⃗
x)
for all unimodular vectors
⃗
x∈
V . Let⃗
x andy be unimodular vectors in V . By⃗
Lemma 2.2, there is an automorphism T∈
SpR(
V)
such that T(⃗
x) = ⃗
y. Thus, we haveσ
T∈
AutGSpR(V)andσ
T:
R⃗
x→
RT(⃗
x) =
R⃗
y.Next, let
⃗
x1, ⃗
x2, ⃗
y1, ⃗
y2be unimodular vectors in V such that Rx⃗
1is adjacent to R⃗
y1and R⃗
x2is adjacent to R⃗
y2. We mayassume that
β(⃗
x1, ⃗
y1) =
1=
β(⃗
x2, ⃗
y2)
. That is,{⃗
x1, ⃗
y1}
and{⃗
x2, ⃗
y2}
are hyperbolic pairs. Again, byLemma 2.2, there existsan automorphism T
∈
SpR(
V)
such that T(⃗
x1) = ⃗
x2and T(⃗
y1) = ⃗
y2. Therefore,σ
T∈
AutGSpR(V)sends Rx⃗
1to R⃗
x2and R⃗
y1to R⃗
y2as desired.Lemma 2.4. Let p be a prime, and let n be a positive integer. Then,Zpnhas stable range one.
Proof. Let
α, β ∈
Zpnbe such that(α, β) =
Zpn. If p|
α
, then p-β
, so p-(α +
1β)
and if p-α
, then p-(α +
0β)
. Hence,Zpnhas stable range one.
Back to the symplectic graph studied in Section1. If p is a prime,
ν
and n are positive integers, R=
Zpn, V=
Z2pνn andβ : (⃗
x, ⃗
y) → ⃗
xK(2ν)⃗
yt for all⃗
x, ⃗
y∈
V , then it follows thatGSpR(V)is isomorphic to Sp(2ν)
pn . Thus,Theorem 2.3andLemma 2.4
show that Sp(p2nν)is vertex transitive and edge transitive when p is odd. We record this result in the following theorem. Theorem 2.5. Let p be an odd prime, and let n and
ν
be positive integers. The symplectic graph Sp(p2nν)is vertex transitive andedge transitive.
Recall from Proposition 2.10 of [2] that the center of SpR
(Z
2pnν)
is{±
I2ν}
, and so we have the quotient group PSpR(Z
2pνn) =
SpR(Z
2νpn
)/{±
I2ν}
, which is called the projective symplectic group of degree 2ν
overZpn. Our final theorem tells us that we mayregard PSpR
(Z
2pνn)
as a subgroup of the automorphism group of the symplectic graph Sp(2ν)
pn .
Theorem 2.6. For any T1and T2in SpR
(Z
2ν
pn
), σ
T1=
σ
T2 if and only if T1= ±
T2, where the induced automorphismσ
T,
T∈
SpR(Z
2pνn)
, on the vertex set of the symplectic graph Sp(2ν)
pn is given by
σ
T:
Zpn⃗
x→
ZpnT(⃗
x)
Proof. It is clear that
σ
T1=
σ
T2if T1= ±
T2. For the opposite direction, suppose thatσ
T1=
σ
T2. That is,ZpnT1(⃗
x) =
ZpnT2(⃗
x)
for all unimodular vectors⃗
x. Thus, byLemma 2.1, for any unimodular vector⃗
x∈
Z2pνn,
T1(⃗
x) = λ
T2(⃗
x)
for someλ ∈
Z×
pn.
In particular, if
{⃗
e1, ⃗
e2, . . . , ⃗
e2ν}
is the standard basis forZ2pnν, we have for i∈ {
1,
2, . . . ,
2ν},
T1(⃗
ei) = λ
iT2(⃗
ei)
for someλ
i∈
Z×
pn. Let i
∈ {
1,
2, . . . , ν}
. Since T1and T2are isometries onZ2pνn,⃗
eiK(2ν)
⃗
eν+t i=
T1(⃗
ei)
K(2ν)T1(⃗
eν+i)
t=
λ
iλ
ν+iT2(⃗
ei)
K(2ν)T2(⃗
eν+i)
t=
λ
iλ
ν+i⃗
eiK(2ν)⃗
etν+i which impliesλ
iλ
ν+i=
1. In addition,⃗
eiK(2ν)
⃗
1t=
T1(⃗
ei)
K(2ν)T1(⃗
1)
t=
λ
iλ
T2(⃗
ei)
K(2ν)T2(⃗
1)
t=
λ
iλ⃗
eiK(2ν)⃗
1t,
and(−⃗
eν+i)
K(2ν)⃗
1t=
T1(−⃗
eν+i)
K(2ν)T1(⃗
1)
t=
λ
ν+iλ
T2(−⃗
eν+i)
K(2ν)T2(⃗
1)
t=
λ
ν+iλ(−⃗
eν+i)
K(2ν)⃗
1t,
where⃗
1= [
1,
1, . . . ,
1]
2νand T1(⃗
1) = λ
T2(⃗
1)
for someλ ∈
Z×
pn. Thus,
λ
1=
λ
2= · · · =
λ
2ν=
λ
−1.
Recall that
λ
iλ
ν+i=
1 for all i∈ {
1,
2, . . . , ν}
, we finally reach T1= ±
T2as desired.Acknowledgments
This work was finished while the first author was visiting the National Center for Theoretical Sciences (NCTS), Mathematics Division, Hsinchu, Taiwan under his exchanged program sponsored by Chulalongkorn University. He expresses his gratitude to NCTS hospitality and academic funding of Chulalongkorn University. The second author would like to thank the Development and Promotion of Science and Technology Talents Project for partially support. Last but not least, we wish to thank the anonymous referees for valuable comments which corrected and improved the quality of the paper.
References
[1] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001.
[2] B. Kirkwood, B.R. McDonald, The symplectic group over a ring with one in its stable range, Pacific J. Math. 92 (1) (1981) 111–125. [3] Z. Tang, Z. Wan, Symplectic graphs and their automorphisms, European J. Combin. 27 (2006) 38–50.
