k-Diameter and Spectral Multiplicity C. Delorme, F.R.K. Chung, University of Paris-Sud University of Pensylvania, 91405 Orsay, Philadelphia, PA 19 104 France USA Email:
[email protected] Email:
[email protected] Patrick Sole CNRS, I3S ESSI, BP 145 Route des Colles 06 903 Sophia Antipolis France Email:
[email protected] May 2, 1997 Abstract
The k-diameter of a graph is the largest pairwise minimum distance of a set of k vertices in , i.e. the best possible distance of a code of size k in . We study the function N (k; ; D) the largest size of a graph of degree at most and k-diameter D. The graphical analogues of the Gilbert bound and of the Hamming bound of coding theory are derived Constructions of large graphs with given degree and k-diameter are given. Eigenvalue upper bounds are derived. Combining sphere packing arguments and eigenvalue bounds new lower bounds on spectral multiplicity are derived.
1 Introduction The diameter of a graph measures how far two distinct points can be; similarly the k-diameter (de nition given below) measures how far k points can be; in other words how good can a code of size k in the graph be. Two problems on the diameter have excited a great deal of attention since the 0 80s: the (; D) graph problem: how large can be a graph of bounded degree and given diameter? [1, 6, 5] nding the best spectral upper bound on the diameter of a graph. [2, 3] This work is an attempt to generalize both philosophies. First, we study a function N(k; ; D) the largest size of a graph of degree at most and given k-diameter D. Observe that this is a very hard problem which comprises as special instances both Moore graphs and perfect codes. We begin a 3-D table collecting the sizes of the largest such graphs. No exact value with > 2 is known so far. Second we derive the natural analogues of the Chung et al. upper bounds [2, 3, 7] on the diameter. Combining the two types of argument we obtain some lower bounds on the spectral multiplicity of a kind which seems to be new. 1
2 De nitions and Notations All graphs considered are nite, connected, with vertex set V , simple, undirected, without loops or multiple edges. The graphical distance d(x; y) between two vertices x and y is the length of a shortest path between x and y. The girth hereby denoted by g is the shortest size of a circuit. A k-code in such a graph with distance d is a set of k( 2) vertices with min (d(xi; xj )) = d: i6=j The k-diameter of say Dk is the largest possible distance a k-code in can have. Note that D2 is the standard diameter. For instance D3 = n in the ternary Hamming graph H(n; 3). The sequence k 7! Dk is nonincreasing: D2 D3 Dk Dk+1 The Moore function is for ; D integers M(; D) := 1 + + ( 1) + + ( 1)D 1: or, in closed form ( 1)D 2 if > 2 M(; D) = 2 2D + 1 if = 2. Its bipartite counterpart Mb (; D) is 2(1 + ( 1) + + ( 1)D 1), or, in closed form Mb (; D) =
2( 1)D 2 if > 2 2
2D
if = 2. A distinct but related quantity is the covering radius r(C) of a code C is r(C) := max min d(v; c): v2V c2C
Denote by T the diagonal matrix indexed by V such that Tx;x is the degree of x 2 V , by A the adjacency matrix and let L = T A. The Laplace operator is then de ned as L := T 1=2LT 1=2 : Let 0 = 0 1 : : : v 1 be the eigenvalues of the Laplace operator arranged in increasing order and normalized in such a way that the whole spectrum ts into [0; 2]. Concretely if the graph is -regular then i = 1 i =, It seems that a where i is the ith eigenvalue (decreasing order) of the adjacency matrix. normalization with 3 Sphere Packings and Coverings 1 + v 1 = 2 is pre3.1 An improved Gilbert bound ferred in theorem 3, The classical Moore bound is related to N(2; ; D) following the de nition of the 2-diameter this one is for N(2; ; D) M(; D): (1) used corollary 2 The analogue of the Gilbert bound is N(k; ; D) (k 1)M(; D); which can be proved by induction on k (equation (1) is the case k = 2). We can also have the following result relating N(k; ; D) to codes with covering radius D 2
q=0 q=1
q=0
q=2 v
w d=2 d=1 c=0 c=1 c=2 c=3 c=4
Figure 1: Explanation of G(; D)
Lemma 1
N(k; ; D) F(k; ; D) (2) where F(k; ; D) denotes the largest size of the vertex set V of a graph containing a (k 1)-code C with minimum distance D + 1, and covering radius at most D. Proof : Let be a graph with degree at most , satisfying Dk D on N(k; ; D) vertices. Consider a (k 1)-code C with minimum distance D + 1. Then we claim that its covering radius is at most D otherwise we would obtain a k-code with distance D +1 contradicting the hypothesis Dk D. 2 We are now in a position to give a small improvement on the Gilbert bound by taking into account graph connectedness and ball intersections. Theorem 1 We claim that a graph with k-diameter D and maximum degree has at most (k 1)M(; D) (k 2)M(; (D 1)=2) vertices if D is odd and (k 1)M(; D) (k 2)Mb (; D=2) if D is even. Proof : If the graph is connected, we can see that if v and w are vertices at distance D + 1, then the number of vertices at distance at most D from w and at least D + 1 from v is at most G(; D) =
D X c=0
( 1)c +
bD= X2c d=1
( 2)
DX2d q=0
( 1)D d 1 q
The explanation of that formula should be clear from gure 1. We note that G(; D) = M(; D) M(; (D 1)=2) if D is odd and G(; D) = M(; D) Mb (; D=2) if D is even. By Lemma 1, considering a k 1-code of distance D + 1 where we specialize one ball of radius D about v and a sequence of k 2 balls each of which meets some previous one. 2 For = 2, the bound is 2D + 1 + (k 2)(D + 1), instead of (k 1)(2D + 1), the exact value is k(D + 1) 1 (and the optimal graph is a cycle).
3.2 The Hamming bound
Let e(D) := D2 1 . Assume Dk g. Then the analogue of the Hamming bound reads v kM(; e(Dk )):
3
Equality corresponds to the case of a perfect code in a graph such that the volume of a ball of radius e(D) is indeed M(; e(D)). This happen for instance if the graph is regular with girth at least 2e(D) + 1. For instance if n = 2m 1 the Hamming codes yield N(2n m; n; 3) = 2n = 2n m (2n + 1); meeting the preceding bound with equality. Similarly perfect Lee metric codes of length n over Fp with p = 2n + 1 yield N(pn 1; 2n; 3) = pn = pn 1(1 + 2n): But the repetition codes of length 2s + 1 yield only (for s > 1) the inequality N(2; 2s + 1; 2s + 1) 22s+1: Observe that the volume of a ball of radius s in the (2s + 1)-hypercube is 22s < M(2s + 1; 2s + 1).
4 Constructions
4.1 Graphs of large girth
A relation between the girth and the k-diameter is g kDk + 1: This bound is of special interest in the case of incidence graphs of generalized polygons. These are graphs with diameter N and girth 2N. In that case we obtain the estimates 2N 1 D N: k k The lower bound is met with equality for N = 3; k = 3 case of the incidence graph of a projective plane PG(2; q). It can be directly veri ed in that case that a line l, along with a pair of points non on l constitutes a 3-code with distance 2, and also that D3 2 because a pair of points or a pair of lines are at a distance at most 2 apart. So, we obtain, for every prime power q the estimates N(3; q + 1; 2) 2(q2 + q + 1); quite close to the upper bound N(3; q + 1; 2) 2(2 + 2q + q2 ) 2: Similarly in that case we have D4 = 2 yielding N(4; q + 1; 2) 2(q2 + q + 1); to be compared to the upper bound N(3; q + 1; 2) 3(2 + 2q + q2 ) 4:
On garde ca? on a N(4; q + 4.2 Bipartite Graphs 1; 2) Let B(; D) denote the largest size of a bipartite graph of degree at most and diameter D. 3(qq + q + 1) Lower bounds for this function are tabulated for 16 and D 10 in [5], and an upper bound ci-apres is Mb (; D). Given a 3-code in such a graph two of its points at least should lie in the same part. This leads to N(3; ; D 1) B(; D) if D is odd: This can be generalized to multipartite graphs by denoting by P(p; ; D) the largest size of a p-partite graph with diameter D and degree at most . For any D = pm + a with 0 < a < p and m integer we have N(p + 1; ; pm) P(p; ; D): 4
Figure 2: Graph showing N(k; 3; 3) 7(k 1)
4.3 Irregular Graphs
For D = 2 and = q + 1, we have some constructions with n = (q2 + q + 1)(k 1) = (2 + 1)(k 1) (take k 1 copies of the quotient of the incidence graph of a projective plane of order q by a polarity and add edges between the vertices of degree q to obtain a connected graph). On the other hand, the modi ed Gilbert bound is (2 1)(k 1) + 2 instead of (2 + 1)(k 1). For q = 2, we have the kind of graphs of gure 2: More generally, if we have a graph with maximum degree and diameter D, with n vertices among which two at least have degree 1, it is easy to prove N(k; ; D) (k 1)n by arranging a path of k 1 copies of G connected by edges. This gives N(k; ; 1) (k 1), to be compared by the upper bound coming from k 1 n , namely N(k; ; 1) 1 + (k 1). For D = 3; 5 the generalized quadrangles and hexagons give also good results: a (s; t)quadrangle Q has (s+1)(st +1) vertices of degree t+1 and (t+1)(st+1) vertices of degree s+1; one of the components of the Kronecker product of Q by itself is regular of degree (t+1)(s+1), has 2(s + 1)(t + 1)(st + 1)2 vertices and admits an obvious polarity, the quotient is a diameter 3 graph with (s+1)(t+1)(st+1)2 vertices of maximum degree = (t+1)(s+1), with (s+1)(t+1)(st+1) vertices having degree st + t + s. Hence N(k; (t + 1)(s + 1); 3) (k 1)(s + 1)(t + 1)(st + 1)2 , provided that there exists a (s; t)-quadrangle. For each m 0, some quadrangles with s = t = 22m+1 have already a polarity and give N(k; s + 1; 3) (k 1)(s + 1)(s2 + 1). For example, 15(k 1) N(k; 3; 3) 18(k 1) + 4. Similarly, the (s; t)-hexagons yield graphs with (s+1)(t+1)(s2 t2 +st+1)2 vertices, maximum degree = (t + 1)(s + 1), with (s + 1)(t + 1)(s2 t2 + st + 1) vertices having degree st + t + s. Hence N(k; (t + 1)(s + 1); 5) (k 1)(s + 1)(t + 1)(s2 t2 + st + 1)2 , provided that there exists a (s; t)-generalized hexagon, and hexagons of order s = t = 32m+1 admitting a polarity give N(k; s + 1; 5) (k 1)(s + 1)(s2 + s + 1).
4.4 Small values
For D = 2, k = 3 and = 3, we have M = 10 and F = 8, the improved Gilbert bound is 18. If there is a vertex v of degree 2, there are at most 7 vertices at distance 2 from v. If w is at distance 3 from v, there are at most 8 vertices at distance 2 from w and at distance at least 3 from v. Thus a graph with maximum degree 3 and 3-diameter 2 having a vertex of degree 2 has at most 15 vertices. Thus there is no graph with 17 vertices, maximum degree 3 and 3-diameter 2. Let us consider now 3-regular connected graphs. Since 18 < M(3; 3) = 22, every vertex of a 3-regular connected graph with 3-diameter 2 lies in a cycle of length < 7, thus the bound 18 cannot be attained. See gure 3. Thus we can state 14 N(3; 3; 2) 16. Similarly, from the improved Gilbert bound for N(4; 3; 2), that is 26, we can see that the girth is at most 7, and this implies N(4; 3; 2) 24 because we can either nd a vertex of degree 2 leading to a bound of 7 + 8 + 8 = 23 or use the results for a cycle of length 6, or start with a cycle of length 7 as shown in gure 4. The proof is easily generalized to N(k; 3; 2) 8(k 1). 5
16 v
17 w
v
17 v
w 16
w
v
w
Figure 3: Explanation of N(3; 3; 2) 16
Figure 4: Explanation of N(4; 3; 2) 24
k 3 3 3 4 4
3 4 3 3 3
D 2 2 4 2 3
Table 1: Small values N(k; ; D) G IG graph 14-16 20 18 PG(2; 2) 26-32 34 32 PG(2; 3) 56-68 72 68 [5] 21-24 30 26 x 4.3 45-58 66 58 x 4.3
The 4th column gives the known bounds, the 5th and 6th ones present the Gilbert and improved Gilbert bounds.
6
5 Spectral bounds
5.1 Main bounds
Recall that the inverse hyperbolic cosine is given by
p cosh 1 (x) = log(x + x2 1);
for x 1. .
Theorem 2 Suppose G is not a complete graph. For X; Y V (G), we have 2
cosh 1 d(X; Y ) 66 6 cosh
Proof : For X V (G), we de ne
q
3
vol X vol Y vol X vol Y 7 : 1 v 1 +1 7 7 v
1
1
Something wrong, if X = Y , this formula gives 0, instead of 1; (3) one should take 1 + bc instead of de introduce vol, introduce
1 if x 2 X, 0 otherwise . 1 If we can show that for some integer t and some polynomial pt (z) of degree t, hT 1=2 Y ; pt (L)(T 1=2 X )i > 0 then there is a path of length at most t joining a vertex in X to a vertex in Y . Therefore we have d(X; Y ) t. Let ai denote the Fourier coecients of T 1=2 X , i.e., X (x) =
Xn
1
T 1=2 X = i=0 ai i ; where the i's are eigenfunctions of L. In particular, we have 1=2 1=2 X a0 = hThT 1=21X; ;TT1=211i i = vol vol G : Similarly, we write Xn 1 T 1=2 Y = i=0 bi i : Suppose we choose p1(z) = v 21z+1 1 and pt(z) = (p1(z))t . Since G is not a complete graph, 1 6= v 1 , and jpt(i )j (1 )t for all i = 1; ; v 1, where = 21 =(v 1 + 1). Therefore, we have
hT 1=2 Y ; pt(L)T 1=2 X i = a0 b0 +
thogonal
because of the possible multiple eigenvalues
X
a0 b0 (1
p ( )a b i>0 tr i i i X 2 X b2 a )t i>0 i i>0 i
X vol Y = vol vol G
Note that
We should add or-
(1
)t
p
vol X vol X vol Y vol Y : vol G 2
X) a2 = kT 1=2 X k2 (vol i>0 i vol G X vol X : = vol vol G
X
7
, not >
If we choose
q
X vol Y log vol vol X vol Y t> log 1 1
we have
p
hT 1=2 Y ; pt(L)T 1=2 X i > 0:
This proves D(G) 1 + log logvol1 1Xvol Y . It can be improved with another choice for pt, namely the normalized Chebychev polynomial such that 2z 1 1 1 1 cosh(tcosh 1 pt (z) = cosh tcosh 1 v 1 1 vol Y vol X
Then for 1 z v 1 we have jpt(z)j cosh(tcosh1 1 ( 1 1 ))
hT 1=2 Y ; pt(L)T 1=2 X i = a0b0 +
X
p ( )a b i>0 t i i i
r
1 X a2 X b2 a0b0 i>0 i i>0 i cosh tcosh 1 1 1 p 1 vol X vol X vol Y vol Y : vol X vol Y = vol G vol G cosh tcosh 1 1 1
Note that
2
X) a2 = kT 1=2 X k2 (vol i>0 i vol G X vol X : = vol vol G
X
If we choose we have
q
X vol Y cosh 1 vol vol X vol Y t 1 1 cosh 1
hT 1=2 Y ; pt(L)T 1=2 X i > 0:
This proves Theorem 2 As an immediate consequence of Theorem 2, we have Corollary 1 Suppose G is a regular graph which is not complete. Then &
2
'
1 D(G) cosh 1 (vv 1 +1) : cosh v 1 11
To generalize Theorem 2 to distances among k subsets of the vertices, we need the following geometric lemma [4] Lemma 2 Let x1; x2; :::xd+2 denote d + 2 arbitrary vectors in d-dimensional Euclidean space. Then there are two of them, say, vi ; vj (i 6= j ) such that hvi ; vj i 0.
8
Theorem 3 Suppose G is not a complete graph. For Xi V (G); i = 0; 1; ; k, we have q
2
cosh 1 6 min d(X ; X ) max i j i6=j i6=j 6 cosh 6
volXi volXj 3 volXi volXj 7 1 1 7 7 1 k
if 1 k v 1 1.
Proof : Let X and Y denote two distinct subsets among the Xi 's. We consider hT 1=2 Y ; (I L)tT 1=2 X i a0 b0 +
1 (1 i )tai bi i=1
Xk
X
ik
(1 k )t jaibi j:
For each Xi ; i = 0; 1; ; k, we consider the vector consisting of the Fourier coecients of the eigenfunctions '1 ; ; 'k 1 in the eigenfunction expansion of Xi . Suppose we de ne a scalar product for two such vectors (a1 ; ; ak 1) and (b1 ; ; bk 1) by Xk 1 (1 i )tai bi : i=1 From Lemma 2, we know that we can choose two of the subsets, say, X and Y with their associated vectors satisfying Xk 1 (1 i )tai bi 0: i=1 Therefore, we have p volX volY X volY volY 1 = 2 t 1 = 2 t hT Y ; (I L) T X i > vol G (1 k ) volX vol vol G and Theorem 3 is proved. 2 We note that the condition 1 k v 1 1 can be eliminated by modifying the 's as follows: Theorem 4 For Xi V (G); i = 0; 1; ; k, we have 2
cosh 1 6 min d(Xi ; Xj ) max i6=j i6=j 6 6 cosh
q
volXi volXj 3 volXi volXj 7 1 v 1 +k 7 7 v
1
k
if k 6= v 1 .
Observing that the denominator is a decreasing function of v 1 2 we obtain the following useful corollary. Corollary 2 For Xi V (G); i = 0; 1; ; k, we have q
2
cosh 1 6 min d(Xi ; Xj ) max i6=j i6=j 6 cosh 6
volXi volXj 3 volXi volXj 7 1 2+k 7 7 2 k
if k 6= v 1 .
5.2 Spectral Multiplicity
We now derive an application of the preceding bounds to estimate on the spectral multiplicity of graphs. Theorem 5 If A1 denotes the multiplicity of 1 for regular graph on v vertices,then A1 v=M(; dcosh 1(v)=cosh 1( 22 + 1 )e): 1 9
Here also we should take pt (normalized Chebychev) and not these powers.
Proof : Consider an A1 -code in the considered graph. Then its minimum distance is at most dcosh 1 (v)=cosh 1 ( 2+ 2 11 )e. We eliminate the A1-diameter by the standard Gilbert argument observing that an upper bound on a ball of radius D in the graph we consider is M(; D). 2 This can be generalized further to obtain bounds on the distribution of the rst m eigenvalues say. Theorem 6 If Ai denotes the multiplicity of the ith distinct eigenvalue for a regular graph on Pm v vertices, and k = j =1 Aj then m X
Aj v=M(; dcosh 1(v)=cosh 1( 22 + k )e): k j =1
More generally if we have a graph G with balls of radius D bounded by a function MG (D) we obtain the following result. Theorem 7 With the above hypothesis, If Ai P denotes the multiplicity of the ith distinct eigenvalue for a regular graph on v vertices, and k = m j =1 Aj then m X
Aj v=MG (dcosh 1 (v)=cosh 1( 22 + k )e): k j =1
For instance for abelian Cayley graphs of degree = k1 + 2k2 (ki =number of generators of order 2) we get from [8] MG (D) = (2D + k)k =k! where k = k1 + k2 .
6 Acknowledgement The third author is indebted to J. Bond and C. Garcia for references [5, 8] and helpful discussions.
References [1] J.C. Bermond, C. Delorme, G. Fahri, Large graphs with given degree and diameter, J. Comb. series B (1984) 32-48. [2] F.R.K. Chung, Diameter and eigenvalues, J. of the AMS 2 (1989) 187-196. [3] F.R.K. Chung, V. Faber, Thomas A. Manteu el, On the diameter of a graph from eigenvalues associated with its Laplacian, SIAM J. of Discrete Math. 7 (1994) 443-457. [4] F.R.K. Chung, A. Grigor'yan, S.-T. Yau, Upper bounds for eigenvalues of the discrete and continuous Laplace operators, Advances in Math. 117 (1996) 165-178. [5] J. Bond, C. Delorme, New large bipartite graphs with given degree and diameter, Ars Comb. 25C (1988) 312-317. [6] C. Delorme, Grands graphes de Degre et Diametre donnes, Europ. J. of Comb. 6 (1985) 291-302. [7] Delorme, C. and Sole, P., Diameter, covering number, covering radius and eigenvalues, European J. Combinatorics 12 (1991), 95-108. [8] C. Garcia, PhD Thesis, Universite de Nice-Sophia Antipolis (1993). [9] F.J. MacWilliams, N.J.A. Sloane,The Theory of Error-Correcting Codes, North Holland (1977). 10
