The spectrum of Platonic graphs over finite fields

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1 Discrete Mathematics The spectrum of Platonic graphs over finite fields Michelle DeDeo a Dominic Lanphier b Marvin Minei c a Department of Mathematics University of North Florida Jacksonville FL USA b Department of Mathematics Western Kentucky University Bowling Green KY 4211 USA c Department of Mathematics University of California Irvine CA USA Received 15 September 25; received in revised form 26 June 26; accepted 2 July 26 Available online 13 October 26 Abstract We give a decomposition theorem for Platonic graphs over finite fields and use this to determine the spectrum of these graphs. We also derive estimates for the isoperimetric numbers of the graphs. 26 Elsevier B.V. All rights reserved. Keywords: Cayley graph; Ramanujan graph; Isoperimetric number 1. Introduction Let p be an odd prime let r 1 and let F q be the finite field with q = p r elements. We consider a generalization of the Platonic graphs studied in [27 9]. In particular using the notation from [7] we define our graphs G q as follows. Definition 1. Let the vertex set of G q be VG q ={α β α β F q αβ = }/ ±1 and let α β be adjacent to γ δ if and only if det α γ β δ =±1. For q = p the graph G q is called the qth Platonic graph. The name comes from the fact that when p = 3or 5 the graphs G p correspond to 1-skeletons of two of the Platonic solids in other words the tetrahedron and the icosahedron as in [2]. More generally G p is the 1-skeleton of a triangulation of the modular curve Xp see [7]. The spectrum Λ G of a graph G is the set of eigenvalues of the adjacency operator see [4]. For a finite k-regular graph G it is known that k Λ G. Further λ Λ G satisfies λ k. Let {G m } m 1 be a family of finite connected k-regular graphs with lim m VG m =. Let λ 1 G m be the largest element in Λ Gm distinct from k. Then it is known that lim inf m λ 1 G m 2 k 1 see [15]. Corresponding author. addresses: mdedeo@unf.edu M. DeDeo dominic.lanphier@wku.edu D. Lanphier mminei@math.uci.edu M. Minei X/$ - see front matter 26 Elsevier B.V. All rights reserved. doi:1.116/j.disc

2 M. DeDeo et al. / Discrete Mathematics A k-regular graph G is called Ramanujan if for all λ Λ G with λ = k we have λ 2 k 1. From the inequality for λ 1 above we see that this estimate is optimal for families of regular graphs of fixed degree. It follows that Ramanujan graphs make the best expanders in the sense of [11]. In this paper we determine the spectrum of G q and thus generalize results of [7]. Theorem 1. Let p>2 be prime and q = p r. The spectrum of G q is q with multiplicity 1 1 with multiplicity q and ± q each with multiplicity q + 1q 3/4. It follows that the graphs G q are Ramanujan. To do this we demonstrate a decomposition theorem for the graphs G q which we also use to estimate their isoperimetric numbers. Let K n denote the complete graph on n vertices and let Kn m denote the complete multigraph on n vertices with m edges connecting each pair of distinct vertices. For α in VG q let H α be the graph defined in Section 3. Thus H α has q + 1 vertices consisting of p r 1 disjoint p-circuits with every vertex in a given circuit adjacent to the vertex α. Theorem 2. The graph G q can be partitioned into q 1/2 disjoint copies of H α with 2q edges joining every pair of H α s. Alternatively G q is the complete multigraph K 2q q 1/2 where each vertex should be viewed as a copy of H α. For an arbitrary graph G and S V G the boundary of S denoted S is the set of all edges having exactly one endpoint in S. The isoperimetric number [12] is defined to be isog = inf S S S where the infimum is over all S V G so that S V G /2. The isoperimetric number also called the Cheeger constant is a measure of the connectedness of G. A set S where isog = S / S is called an isoperimetric set for G. The isoperimetric number has applications to combinatorics and computer science. For example if a graph G is viewed as a communications network then a large isog means that information is transmitted easily throughout the network. A small isoperimetric number means that communication can be easily disrupted. As a consequence of Theorems 1 and 2 we obtain the following bounds for the isoperimetric numbers isog q of G q. Corollary 1. We have qq 1 q q if q 1 mod isog q 2q + 1 q if q 3 mod 4. 2 In the next section we define a Cayley graph Gq of PSL 2 F q and consider a graph G q obtained from the quotient of PSL 2 F q by the unipotent subgroup. We relate G q to G q in order to prove Theorem 2. We then demonstrate bounds for the isoperimetric number of G q using the methods of [89]. In the last section we use the decomposition theorem to determine the spectrum of G q. In [1] Li and Meemark determined the spectrum of Cayley graphs on PGL 2 F q modulo either the unipotent subgroup the split torus or the nonsplit torus of G. They use the Kirillov models of the representations of PGL 2 F q. This is a different approach from the one used here or in [6] as[1] relies primarily on representation theory rather than an analysis of the graph structure to determine the spectrum. Note that in this paper character sums are not needed to evaluate the eigenvalues. 2. A Cayley graph of the projective linear group The Cayley graph of a group Γ with respect to a symmetric generating set Ω is the graph with vertex set Γ and with two vertices γ 1 γ 2 joined by an edge if we have γ 1 = αγ 2 for some α Ω. Cayley graphs are vertex transitive and

3 176 M. DeDeo et al. / Discrete Mathematics Ω -regular. In this section we define a Cayley graph of Γ q = PSL 2 F q for q = p r and p>2 a prime. We then study a quotient graph of the Cayley graph. For q 1 mod 4 there exists some ω F q so that ω 2 = 1. From [3] for appropriately chosen x F q the set Ω q = { 1 ω ω 1 ± x is a generating set for Γ q.forq 3 mod 4 and q>11 from [3] PSL 2 F q is generated by { } a b ω 1 b a ω 1 } where a 2 + b 2 = 1 ω F q and ω = ±1. Since q 3 mod 4 both a and b are nonzero. Since a b 1 a 1 b a 1 b = 1 b a 1 a we have that PSL 2 F q is generated by { ω Ω q = 1 ω 1 ω 1 ω } 1 ± x for appropriately chosen x F q. We take Ω q to be the symmetric generating set for PSL 2 F q for q 1 3 mod 4 where ω 1 = ω in the former case. Thus Ω q =4 for q 1 mod 4 and Ω q =5 for q 3 mod 4. Note that for q 3 mod 4 and q 11 q is a prime. In that case Theorems 1 and 2 follow from [78]. Let Gq be the Cayley graph of Γ q with respect to Ω q. Note that V Gq =qq 2 1/2. Let N = { 1 x x F q }. Thus N =q. Let Γ q = N\Γ q and note that Γ q { α β α β F q α β = }/ ±1 which is the vertex set of G q. Let G q denote the quotient graph N\Gq. That is G q is the multigraph whose vertices are given by the cosets Γ q with distinct cosets Nγ 1 and Nγ 2 joined by as many edges as there are edges in Gq of the form α 1 α 2 where α j Nγ j. Note that Γ q is not a group and thus the graphs G q are not themselves Cayley graphs. Also note that the vertex set of G q is Γ q. Lemma 1. Let α β γ δ G q. Then α β is adjacent to γ δ if and only if α β det =±1 ±ω or ± ω 1. γ δ Proof. Left multiplication of g Γ q by elements of N preserves the bottom row of g. So g G q is adjacent to precisely those elements attained by left multiplication by γ Ω q with γ / N. Left multiplication of g by 1 multiplies the top row of g by 1 and switches the rows. Multiplication by ω 1 ω multiplies the top row of g by ω and switches the rows and similarly multiplication by ω ω 1 multiplies the top row of g by ω 1 and switches the rows. So if g = α β γ δ Γ q with determinant equal to 1 and ξg = α β γ δ for ξ Ω q N then γ δ = α β ω α βor ω 1 αβ. Therefore we must have det γ δ γ δ =±1 ±ω or ± ω 1.

4 If det α γ ω 1 β δ =±1 ±ω or ±ω 1 then εα γ or ω 1 ω ω M. DeDeo et al. / Discrete Mathematics εβ δ Γ q for some ε {±1 ±ω ±ω 1 }. Thus multiplication by 1 takes γ δ to α β inside Γ q. It follows that these vertices are adjacent in G q. Note that VG q =q 2 1/2. Two edges incident with g Gq arise by multiplication by the two elements of Ω q that are in N. The remaining Ω q 2 edges come from cosets Nx Ny with xy / N. Further any vertex in the graph G q corresponds to q vertices in Gq. It follows that the regularity of G q is Ω q 2 and EG q = Ω q 2qq The decomposition theorem 1 =±1 ±ω or ±ω 1 from Lemma 1. If the Let α β and γ δ be adjacent vertices in G q and thus det α β γ δ determinant is ±1 then call the edge incident with these vertices a 1-edge and otherwise call the edge an ω-edge. Note that G q is obtained from G q simply by removing all of the ω-edges. Further it is easy to see that the number of ω-edges incident with a vertex in G q is Ω q 3 times the number of 1-edges incident with that vertex. It follows that G q is q-regular. Let α F q and define V α = { α α 1 β β F q }. Note that V α =q + 1. If α F q α = ±α then it is easy to see that V α V α =. Otherwise V α = V α. It is also easy to see that any vertex in G q lies in V α for some α F q and therefore VG q can be partitioned into disjoint V α s. In particular VG q can be partitioned into VG q / V α =q 1/2 many copies of V α. Define H α to be the subgraph of G q induced by V α.any α 1 β H α belongs to a p-circuit in H α given by the following sequence of adjacent vertices: {α 1 β α 1 β + α α 1 β + 2α...α 1 β + p 1α α 1 β}. Further any two such circuits share a common vertex α 1 β + jα = α 1 β + kα if and only if β β = k jα and this occurs if and only if such circuits are identical. Thus there are p r 1 disjoint p-circuits in a given H α and each vertex in such a circuit is adjacent to α. We refer to the vertex α as the center of H α.as α 1 β H α is adjacent to α 1 β H α if and only if β =±α + β then α 1 β is adjacent to only two other vertices in H α other than the center. This accounts for all the possible edges in H α and so EH α =2q for any α F q. Suppose H α = H α and α 1 β H α is adjacent to α 1 β H α. Then det α 1 β α 1 =±1 x and the number of solutions for x to this equation is independent of the choice of α F q. The number of 1-edges connecting H α to H α is therefore independent of α and α where H α = H α. Since H α contains 2q edges and there are q 1/2 copies of H α in G qwehave α F H α contains qq 1 edges. As VG q =q 2 1/2 and G q q is q-regular EG q =qq 2 1/4. Thus there are qq 2 1/4 qq 1 = qq 1q 3/4 edges joining different H α s. As there are q 1/2 2 different ordered pairs α α that give distinct H α s and H α s the number of edges joining a given H α to a distinct H α is qq 1q 3/4 = 2q. q 1/2 2 This gives the decomposition as stated in Theorem 2.

5 178 M. DeDeo et al. / Discrete Mathematics The isoperimetric number In this section we use the decomposition theorem to estimate the isoperimetric number of G q as stated in Corollary 1. In addition we estimate the isoperimetric numbers of the related graphs G q and Gq. According to [12] the isoperimetric number for the complete graph K n is { n/2 n even isok n = n + 1/2 n odd. As a consequence of the decomposition theorem for our set S we can take q 1/4 copies of H α if q 1 mod 4 and q + 1/4 copies of H α if q 3 mod 4. In the former case we have q 1 S =q q 1 2 S =2q 4 and we get isog q S S = qq 1 2q + 1. The q 3 mod 4 case follows similarly and we get isog q q/2. We can follow the arguments from [29] to obtain lower bounds for isog q independent of Section 5. Thus we do not need to determine the spectrum in order to demonstrate these graphs are expanders. As this allows us to obtain lower bounds for the isoperimetric numbers of the related graphs G q we include the argument. The proof for the following lemma is the same as in [2]. Lemma 2. If α β α β G q with det α β α β = then there are exactly two paths of length two joining α β to α β. Partition the vertices of G q into sets S 1 and S 2 with S 1 S 2. From Lemma 2 it follows that at least one edge from each of these paths must be cut for every pair v 1 S 1 and v 2 S 2 where v 1 and v 2 are not multiples of each other. Since any v G q has q 1 nonzero multiples the number of such pairs is at least S 1 S 2 q + 1.As G q is q-regular each edge can lie in no more than 2q 1 different paths of length 2. Thus we have S 1 2 S 1 S 2 q + 1 2q 1 and since S 2 q 2 1/4 we get S 1 S 1 S 2 q + 1 q 1 q 3. 4 This gives the lower bound q 3/4 isog q. It follows in this elementary way that these are expander graphs in the sense of [11]. In a similar manner we can apply a version of Lemma 2 to G q and get the bounds qq 1 q 1q 3 if q 1 mod 4 isog q + 1 q 2q 1 3q if q 3 mod 4. 2 By the construction of G q each vertex of G q corresponds to q vertices of Gq. Further S VG q corresponds to a set S V Gq containing q S vertices and it is easy to see that for such sets S = S.

6 M. DeDeo et al. / Discrete Mathematics Therefore the estimate isog q S / S implies the estimate isogq S /q S. Consequently we get isogq q 1/q + 1 for q 1 mod 4 and isogq 3/2 for q 3 mod 4. However it does not follow that a lower bound for the isoperimetric number of a quotient graph implies a lower bound for the isoperimetric number of the larger graph. In general we expect the isoperimetric numbers of the two graphs to be related but an isoperimetric set for Gq may possibly be constructed that cuts through the subgraphs of Gq associated to the vertices in the quotient graph G q. This may give a smaller isoperimetric number than isog q/q. Using the results of the next section we can get an improved lower bound for isog q. Forλ 1 = k the second largest eigenvalue of the adjacency matrix of G qwehave q 2 λ 1 2 isog q. See [1] or [5]. From the results of Section 5 we have λ 1 = q and so obtain q 2 q 2 isog q which gives Corollary The spectrum of Platonic graphs over F q Let S q be the complex vector space of functions s : VG q C equipped with the canonical basis {e v } v VG q where e v w = δ vw and δ vw is the Kronecker delta function. The adjacency operator A q : S q S q is the linear transformation defined by A q sv = sw. w is adjacent to v The spectrum of G q denoted Λ q is the set of eigenvalues of A q.asa q is a symmetric matrix with respect to the basis {e v } v VG q then Λ q R. Further since G q is q-regular we know that Λ q [ qq] and the eigenvalue q occurs with multiplicity 1. Note that Γ q acts on S q and that this action commutes with A q. Let ν be a generator of F q let μ = ν ν 1 and let λ Γ q be of order q + 1/2. Then the conjugacy classes of Γ q are those of 1 2 = 1 b 1 = 1 1 b 2 = 1 ν μ k for 1 k q 5/4 and λ l for 1 l q 1/4. There are five types of representations of Γ q see [13] for example. There is the trivial representation 1 and the Steinberg representation ψ St of degree q. For even characters θ : F q C there are the principal series representations χ θ of degree q + 1. These occur with multiplicity 2. There are q 5/4 isomorphism classes of this representation for q 1 mod 4 and q 3/4 for q 3 mod 4. There are the q 1/4 isomorphism classes of the discrete series representations j of degree q 1. Finally for q 1 mod 4 there are the two split principal series ξ ± of degree q + 1/2. Let S r be the irreducible submodule of S q generated by the isomorphism classes of the representation r. The values of the relevant irreducible characters of Γ q on representatives of the isomorphism classes are given by the following table: 1 2 b 1 b 2 μ κ λ l ψ St q 1 χ θ q θν κ + θν κ ξ ± q + 1/2 1 ± 2/2 1 2/2 1 κ Let χ be the character of Γ q acting on S q. Then trivially χ1 2 = q 2 1/2. Since the stabilizer in Γ q of 1 VG q is isomorphic to N and any element in the stabilizer fixes only the associates of 1 it follows that

7 18 M. DeDeo et al. / Discrete Mathematics χb 1 = χb 2 = q 1/2. From the definition of Γ q no other elements of Γ q stabilize a vertex. Therefore χc = for c not in the classes of 1 2 b 1 and b 2. From these values of χ and the character table we can easily determine that the representations 1 ψ St and χ θ occur with multiplicities 1 1 and 2 respectively. Further for q 1 mod 4 the representations ξ ± occur with multiplicity 1. By dimension counting these are the only occurring representations. It follows that S q decomposes into the direct sum of the submodules S 1 S ψst S χθ and S ξ±. For ε F q let s v S q be defined by s v εv = q and s v w = 1 for all other vertices. For v the center vertex of H α and v not associate to v we can easily calculate from Theorem 2 that A q s v v = 1. Therefore we can see that for any v VG q { A q s v v q for v = = εv 1 otherwise = s v v. Thus s v is an eigenvector with eigenvalue 1. Let M be the submodule of S q generated by all the s v s above. We see that s v = s εv for all v VG q and ε F q. Since v VG q s v = we have dim M q. Further since the intersection of M with the submodule generated by s w with w adjacent to a fixed v has dimension q it follows that dim M = q. As the submodules S r are irreducible we must have M = S ψst.asγ q and A q commute and S ψst is irreducible then A q acts on S ψst by a scalar and it follow that S ψst is a q-dimensional eigenspace with eigenvalue 1. Let s ± θv εv =±θε q if v = v s ± w = θε if w is adjacent to v and otherwise. From Theorem 2 we can θv compute A q s ± θv εv = s ± θv w w is adjacent to εv = θεq =± qs ± θv εv and for w εv we have A q s ± θv w = s± θv v + w is adjacent to w w =v =±θε q + =± qs ± θv w. ε F q θε s ± θv w Thus s ± θv is an eigenvector of A q with eigenvalue ± q. Trivially s ± θv / S 1 or S ψst as they have different eigenvalues. Therefore we must have s ± θv S χ θ ors ξ ± if θ is quadratic. For any even character θ : F q C the above computations show that there are eigenvectors v 1 and v 2 so that A q v i = 1 i q. Further Γq acts on v i with the character θ.ass χθ and S ξ± are irreducible then the orbit of v i Γ q v i must either be a copy of S χθ S ξ± or{}. But as v i Γ q v i then Γ q v i = {}. AsΓ q commutes with A q then A q acts on Γ q v i by a scalar. As Γ q v 1 and Γ q v 2 are the same dimension it follows that Γ q v i are q + 1-dimensional subspaces of S q. That is S χθ and S ξ± are eigenspaces. As Γ q v 1 Γ q v 2 ={} since they are different eigenspaces we have two copies of S χθ in S q and a copy each of S ξ± for quadratic θ as this applies to all the principal series representations. Counting dimensions of the relevant representations we get all of S q. Thus we have completely determined the spectrum of A q and this gives Theorem 1. Acknowledgment The authors thank the referees and Jason Rosenhouse for helpful comments.

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