Automorphisms of the at4(6, 6, 3)-graph and its strongly-regular graphs Текст научной статьи по специальности «Математика»

УДК 519.17+512.54

Automorphisms of the AT4(6,6,3)-graph and its Strongly-regular Graphs

Konstantin S. Efimov*

Ural Federal University Mira, 19, Yekaterinburg, 620000

Ural State University of Economics 8 marta, 62, Yekaterinburg, 620144

Russia

Aleksandr A. MakhneV

N.N.Krasovsky Institute of Mathematics and Mechanics S.Kovalevskoy, 4, Yekaterinburg, 620990

Ural Federal University Mira, 19, Yekaterinburg, 620000 Russia

Received 02.11.2016, received in revised form 10.12.2016, accepted 20.02.2017 Koolen and Jurisich defined class of AT4-graphs (tight antipodal graph of diameter 4). Among these graphs available graph with intersection array {288, 245,48,1; 1, 24, 245, 288} on v = 1 + 288 + 2940 + 576 + 2 = 3807 vertices. Antipodal quotient of this graph is strongly regular graph with parameters (1269, 288,42, 72). Both these graphs are locally pseudo GQ(7, 5)-graphs. In this paper we find possible automorphisms of these graphs. In particular, group of automorphisms of distance-regular graph with intersection array {288, 245,48,1;1, 24, 245, 288} acts intransitive on the set of its antipodal classes.

Keywords: distance-regular graph, strongly-regular graph, automorphism of the graph. DOI: 10.17516/1997-1397-2017-10-3-271-280.

1. Introduction and preliminaries

We consider undirected graphs without loops and multiple edges. Given a vertex a in a graph r, we denote by ^(a) the subgraph induced by r on the set of all vertices, that are at a distance i from a. The subgraph [a] = ri(a) is called the neighbourhood of the vertex a. Let r(a) = r1(a), aL = {a} U r(a). If graph r is fixed, then instead of T(a) we write [a]. For the set of vertices X of graph r through X^ denote .

Let F is some class of graphs. Graph r is called locally F-graph, if [a] lies in F for each vertex a of graph r. If the class F is composed of graphs, isomorphic to a certain graph A, then graph r is called locally A-graph.

Number of verices in neighbourhood of vertex is called degree of a vertex. Graph r is called regular with degree k, if the degree of any vertex a in r is k. Graph r is called edge-regular with parameters (v,k,X), if it has v vertices, and it is regular with valency k, and each edge

* konstantin.s.efimov@gmail.com 1 makhnev@imm.uran.ru © Siberian Federal University. All rights reserved

lies in A triangles. Graph r is amply regular graph with parameters (v,k,A,^), if it is edge-regular with appropriate parameters and [a] n [6] contains n vertices for any two vertices a, b at distance 2 in r. Amply regular graph is called strongly regular graph, if it has diameter 2. Let Kmi,...,mn be a complete multipartite graph {Mi,... ,Mn} with cocliques Mi of order mi. If mi = • • • = mn = m, then this graph is denoted by Knxm.

Koolen and Jurisich [1] defined class of AT4-graphs (tight antipodal graph of diameter 4). Among these graphs available locally pseudo GQ(7,5)-graph with intersection array {288, 245, 48,1; 1, 24, 245, 288}, corresponding to AT4(6, 6, 3)-graph.

Let r be a distance-regular graph with intersection array {288,245,48,1; 1,24,245,288}. Then antipodal quotient r is strongly regular graph with parameters (1269,288,42,72). Both these graphs are locally pseudo GQ(7, 5)-graphs. In this paper we have found possible automorphisms of these graphs.

Theorem 1.1. Let r be a strongly regular graph with parameters (288, 42, 6, 6), G = Aut(r), g be an elment in G with prime order p and Q = Fix(g). Then n(G) C {2, 3, 5, 7} and one of the following statements holds:

(1) Q is empty graph and either p = 2, ai(g) = 24s, or p = 3, ai(g) = 36t;

(2) Q is n-clique and either p = 7, n =1, ai(g) = 84s + 42 or n = 8, ai(g) = 84s, or p = 5, n = 3, ai(g) = 60s + 30 or n = 8, ai(g) = 60s;

(3) Q is l-coclique and either p = 2, l is even, 6 ^ l ^ 36 and ai(g) = 24t — 6l, or p = 3, l is divided by 3, 3 ^ l ^ 36 and ai (g) = 36t — 6l;

(4) Q contains geodesic 2-path, Q does not have vertices of degree 42 and either

(i) p ^ 5 and if p = 5, then |Q| =5t + 3, t = 2, 3,..., 9, or

(ii) p = 3, |Q| = 3s, s = 2, 3,..., 16, and in the case s = 2 Q is complete bipartite graph K33, or

(iii) p = 2, |Q| = 2l, l = 2, 3,..., 24.

Theorem 1.2. Let r be a strongly regular graph with parameters (1269, 288, 42, 72), in which neighbourhoods of vertices are strongly regular graphs with parameters (288,42, 6, 6), G = Aut(r), g be an elment in G with prime order p and Q = Fix(g). Then n(G) C {2, 3, 5, 7, 47} and one of the following statements holds:

(1) Q is empty graph and either p = 47 and ai(g) = 47 • 6, or p = 3 and ai(g) = 126l + 72;

(2) if [a] C Q, then ai(g) = 0, a^ = Q and p = 2;

(3) Q is n-clique and eitherp = 2, n =1, ai(g) = 84t+36, orp = 5, n = 4, ai(g) = 210s+180 or n = 9, ai(g) = 210s, or p = 7, n = 2, ai(g) = 294t + 42 or n = 9, ai(g) = 294t + 84;

(4) Q is l-coclique and either p = 2, l is odd, ai(g) = 84m + 6l + 282, or p = 3, l divided by 3, ai(g) = 126m + 6l + 324;

(5) Q contains geodesic 2-path and either

(i) p = 5, |Q| = 5l + 4, l < 64, for e £ Q we have |Q(e)| = 5t + 3, t = 2,3,..., 9, ai (g) = 210m + 30l + 180, or

(ii) p = 3, |Q| = 3n, n < 108, for e £ Q we have |Q(e)| = 3t, t = 1, 2,..., 8, ai(g) = 18n + 126l + 324, or

(iii) p = 2, |Q| = 2l + 1, l < 161, for e £ Q we have |Q(e)| = 2n, n < 12, ai(g) = 12l + 84s + 288.

Corollary 1.1. Strongly regular graph with parameters (1269, 288, 42, 72), in which neighbourhoods of vertices are strongly-regular graphs with parameters (288, 42, 6, 6), is not vertex transitive.

Theorem 1.3. Let r be a distance-regular graph with intersection array {288,245,48,1; 1, 24, 245, 288}, G = Aut(r), g be an elment in G with prime order p and Q = Fix(g). Then n(G) C {2, 3, 5, 7,47} and one of the following statements holds:

(1) Q is empty graph and either

(i) p = 3, a4(g) = v, or a4(g) divided by 9, a1(g) = 126/ + 234 + 2a4(g) and a3(g) = 252l - 18 - 2a4(g), l < 3, or

(ii) p = 47, a4(g) = 0, alphai(g) = 6 • 47 and alpha^(g) = 12 • 47;

(2) Q is an union of 3 isolated n-cliques and either p = 5, n = 4, 9, or p = 7, n = 2, 9;

(3) p = 3, Q is l-coclique or it contains geodesic 2-path;

(4) p = 2, Q is n-clique, l-coclique or it contains geodesic 2-path.

Corollary 1.2. The group of automorphisms of distance-regular graph with intersection array {288, 245,48,1; 1, 24, 245, 288} acts intransitive on the set of its antipodal classes.

2. Preliminary results

In this section are some of the preliminary results, used in the proofs of Theorems.

Lemma 2.1 ( [2]). Let r be a strongly regular draph with parameters (v,k,X, p) and with non-principal eigenvalues r,s, s < 0. If A is induced regular subgraph from r of degree d on w vertices, then

w(k — d)

s

< d----- < r,

and one of the inequalities reached if and only if when each vertex from r — A is adjacent to exactly w(k — d)/(v — w) vertices from A.

The proof of Theorems use Higmen's method for investigation automorphisms of distance-regular graphs, represented in third chapter in Cameron's monograph [3]. Let r be a distance-regular graph of diameter d with v vertices. Then we have the symmetric association scheme (X, R) with d classes, where X is the set of vertices of r and Ri = {(u, w) e X2 | d(u, w) = i}. For vertex u e X set ki = |Fj(u)|. Let Ai be the adjacency matrix of the graph Tj. Then AiAj = PijAi for some integer numbers pj > 0, which are called the intersection numbers. Note that pjj = |ri(u) fl Tj (w)| for every vertices u, w with d(u, w) = l.

Let Pi be the matrix in which in the (j,l) entry there is pj. Then the eigenvalues k = pi(0),... ,pi(d) of the matrix Pi are eigenvalues of r with multiplicities m0 = !,..., md. Note that the matrix Pi is the value of some integer polinom of P1, so the ordering of eigenvalues of the matrix P1 gives the ordering of eigenvalues of Pi. The matrices P and Q with (i,j) entry pj (i) and Qji = mjpi(j)/ki are called the first and the second eigenmatrix of r and PQ = QP = vI, where I is the identity matrix of order d +1. Let uj and wj be the left and the right eigenvectors of matrix P1 affording eigenvalue p1(j) and having the first coordinate 1. Then the multiplicity mj of the eigenvalue p1(j) is equal v/(uj,wj). In fact, wj are the columns of the matrix P and mj uj are the rows of the matrix Q.

The permutation representation of the group G = Aut(r) on the vertex set of r naturally gives the matrix representation ^ of G in GL(v, C). The space Cv is the orthogonal direct sum

v — w

of the eigenspaces W0, Wi,..., Wd of the adjacent matrix A = Ai of r. For every g £ G we have

0(g) A = A0(g), so the subspace Wi is 0(G)-invariant. Let xi be a character of the representation

d

0Wi. Then for g £ G we obtain xi(g) = v_i Qijaj(g), where a.j(g) is the numbers of vertices

j=o

x of X such that d(x, xg) = j.

Lemma 2.2 ( [4], Lemmas 1-2). Let r be a distance-regular graph with i-th integer eigenvalue di, 0 be a monomial representation of a group G = Aut(r) to the group of linear transformations of the space V = Cv, \i be a character of projection 0 to subspace Wi of dimensionality mi, generated by the eigenvectors of the adjacency matrix of the graph Gamma, corresponding to thetai. Then the following statements hold:

(1) if Q is rational matrix, then for element g from G we have ai(g) = ai(gl) for every l, relatively prime with |g|;

(2) if g is element from G with prime order p, then p divides mi — xi(g);

(3) if g is element from G with order p, p is prime number, then p2 divides mi — xi(gp).

Lemma 2.3 ( [5], Theorem 3.2). Let r be a strongly-regular graph with parameters (v,k,A, and with eigenvalues k,r, —m. If g is automorphism of r and Q = Fix(g), then |Q| ^ v • max{A, n}/(k — r).

3. Automorphisms of a graph with parameters (288,42,6,6)

In this section we assume, that r is strongly-regular graph with parameters (288,42, 6,6) and with spectrum 42i, 6i40, —6i47, G = Aut(T), g is element of G with prime order p Q = Fix(g). By Lemma 2.3 we have |Q| < 288 • 6/36 = 48.

Lemma 3.1. Let xi be a character of representation 0 on subspace of dimension 140. Then

(1) Xi(g) = (6ao(g) + ai(g))/12 — 4 and xi(g) — 140 is divided by p;

(2) if Q is empty graph, then either p = 2, ai(g) = 24s, or p = 3, ai(g) = 36t;

(3) if Q is n-clique, then either p = 7, n =1, ai(g) = 84s + 42 or n = 8, ai(g) = 84s, or p = 5, n = 3, ai(g) = 60s + 30 or n = 8, ai(g) = 60s;

(4) if Q is l-coclique,then either p = 2, l is even, 6 ^ l ^ 36 and ai(g) = 24t — 6l, or p = 3, l is divided by 3, 3 ^ l ^ 36 u ai(g) = 36t — 6l.

Proof. We have

1 1 1

Q= 140 20 —4

1 147 21 3

Therefore xi(g) = 1/72(35ao(g) + 5ai(g) — a2(g)). As a2(g) = v — ao(g) — ai(g), then Xi(g) = (6a0(g) + ai(g))/12 — 4. Finally, xi(g) — 140 is divided by p by Lemma 2.2.

Let Q be an ampty graph. Then p £ {2, 3}. If p = 2, then xi(g) is even, so ai(g) = 24s. If p = 3, then ai(g) = 36t.

Let Q be a n-clique. By the Hoffman boundary we have maximal order of clique in r is not any more than n ^ 1 + k/m = 8. If n =1, then p divides 140 and 147, so p = 7, Xi(g) = (6 + ai(g))/12 — 4 and ai(g) = 84s + 42.

If n > 2, then p divides 8 — n, 35 and 210, so either p = 5, n = 3, xi(g) = (18 + ai(g))/12 — 4 and ai(g) = 60s + 30 or n = 8, xi(g) = ai(g)/12 and ai(g) = 60s, or p = 7, n = 8, xi(g) = ai(g)/12 and ai(g) = 84s.

Let Q be a l-coclique, l > 2. By the Hoffman boundary for cocliques, we have l < vm/(k + m) = 36. Further, p divides 6 and 210 — l, so p = 2, 3. If p = 2, then l is even, any vertex in Г — Q is adjacent to even number of vertices in Q, l > 6, xi(ff) = (6l + «i(g))/12 — 4 and ai(g) = 24t — 6l. If p = 3, then l divided by 3 and a1(g) = 36t — 6l. In case l = 36 any vertex in Г — Q is adjacent to exactly 6 vertices in Q, so a1(g) = 0.

Let Q contains an edge and Q be an union of isolated cliques. Then p divides 35 and 36, contradiction. □

Lemma 3.2. Let Q contains geodesic 2-path b, a, c. Then

(1) Г does not contain its own strongly regular subgraph Д with parameters (v',k', 6, 6);

(2) Q does not contain vertices of degree 42;

(3) p < 5 and if p = 5, then |Q| = 5t + 3, t = 2, 3,..., 9;

(4) if p = 3, then |Q| = 3s, s = 2, 3,..., 16, and in the case s = 2 Q is complete bipartite graph Кз,з;

(5) if p = 2, then |Q| = 2l, l = 2, 3,..., 24, and in the case l = 2 Q is quadrangle.

Proof. Assume that Г contains its own strongly regular subgraph Д with parameters (v', k', 6, 6). Then n2 = 4(k' — 6), n = 2u, k' = u2 + 6 and Д has nonprincipal eigenvalues u, —u. Note that each vertex in Г — Д is adjacent to at most one vertex of Д. Further, the multiplicity of u is f = (u — 1)(u2 + 6)(u2 + u + 6)/(12u), so either u = 2, f = 5, k' =10 and v' = 16, or u = 3, f = 15, k' = 15 and v' = 36. In any case, a vertex of Г — Д is adjacent to more than one vertex of Д, contradiction.

If p ^ 7, then Q is strongly-regular graph with parameters (v', k', 6, 6), contradiction. If Q contains vertex a of degree 42, than for each vertex u £ Г — a^ subgraph [u] П Q contains 6 vertices in [a]. For u £ Q we have [u] П Q = [u] П [a]. If b £ Q — a^, than [b] contained in Q. This contradicts the fact that each vertex of Г — (a^ U Q) is adjacent to 6 vertices in [u] П [a]. So, |Q| = 43 and a1(g) = 0 (otherwise the vertex u, adjacent to ug, the subgraph [u] П [ug ] contains ug and 6 vertices of Q, contradiction). But then x1(g) = 258/12 — 4, contradiction.

Let p = 5. Then Xq,/q £ {1, 6}, |Q| = 5t + 3, t < 9 and degrees of vertices in Q are 2,7,..., 37. If t = 1, then Q = a^ for vertex a of degree 7 of Q and Q contains even number of vertices of degree 7. If Q contains two vertices a, b of degree 7, then Q — {a, b} is coclique. This contradicts the fact that £ {1, 6}. So, Q is regular graph of valency 2 with Xq = = 1, contradiction.

Let p = 3. Then Xq,/q £ {0, 3,6}, |Q| = 3s, s ^ 16 and degrees of vertices in Q are 0,3,..., 39. If s = 2, then Q is complete bipartite graph K3 3.

Let p = 2. Then Xq,/q £ {0, 2,4,6}, |Q| = 2l, l ^ 24 and degrees of vertices in Q are 0,2,..., 40. If |Q| = 4, then Q is quadrangle. □

Lemmas 3.1, 3.2 imply the Theorem 1.1.

4. Automorphisms of graph with parameters (1269,288,42,72)

In this section we assume, that Г is strongly-regular graph with parameters (1269, 288,42, 72) and with spectrum 2881,61080, —36188, in which neighborhoods of vertices are strongly regular with parameters (288,42,6, 6), G = Aut^), g is elment of G with prime order p and Q = Fix(g). By Lemma 2.3 we have |Q| < 1269 • 72/282 = 324.

Lemma 4.1. Let x2 be a character of representation ф on subspace of dimension 188. Then

(1) X2(g) = (6ao(g) — ai(g))/42 + 47/7 and X2(g) — 188 divided by p;

(2) if Q is empty graph, then either p = 47 and ai (g) =47 • 6, or p = 3 and ai(g) = 126l + 72;

(3) if [a] C Q, then ai(g) = 0, a^ = Q and p = 2;

(4) if Q is n-clique, then either p = 2, n =1, ai(g) = 84t + 36, or p = 5, n = 4, ai(g) = 210s+180 or n = 9, ai(g) = 210s, or p = 7, n = 2, ai(g) = 294t+42 or n = 9, ai(g) = 294t+84;

(5) if Q is l-coclique, then either p = 2, l is odd, ai(g) = 84m + 6l + 282, or p = 3, l divided by 3, ai(g) = 126m + 6l + 324.

Proof. For i > 1 we denote ai(g) = pwi. Than we have

So X2(g) = 1/27(4ao(g) - a1(g)/2 + a2(g)/7). As a2(g) = v - a0(g) - a1(g), then X2(g) = (6a0(g) - a1(g))/42 + 47/7. Finally, x2(g) - 188 divided by p by Lemma 2.2.

Let Q be an empty graph. Then p G {3,47}. If p = 47, then x2(g) = -a1(g)/42 + 47/7, so a1 (g) =47 • 6. If p = 3, then x2(g) = (-w1 +94)/14 and a1(g) = 126/ + 72.

Now, in view of Theorem 1.1, we have n(G) C {2, 3, 5, 7, 47}.

If [a] c Q, then for each vertex u G r - Q subgraph [u] n Q contains 72 vertices of [a], so each (g)-orbit on r-Q is coclique and a1 (g) = 0. Further, aL = Q, x2(g) -188 = -140 divided by p, so p = 2, 5, 7. Note than any vertex u G r - Q is sdjacent to at most one vertex of each (g)-orbit of lenght p. There are 140 orbits of length 7 and 196 orbits of length 5, which contradicts the fact that 72 + 139 and 72 + 195 is less than 288. So, p = 2.

Let Q be a n-clique. By the Hoffman boundary we have maximal order of clique is not any more than n ^ 1 + k/m = 9. If n =1, then p divides 288 and 980, so p = 2, number X2(g) = (144 - w1)/21 is even and so a1(g) = 84t + 36. If n > 2, then p divides 44 - n and 245, so either p = 5, n = 4, a1(g) = 210s + 180 or n = 9, a1(g) = 210s, or p = 7, n = 2, X2(g) = 9 - 7w1/42, a1(g) = 294t + 42 or n = 9, x2(g) = 8 - a1(g)/42 and a1(g) = 294t + 84.

Let Q be a l-coclique, l > 2. By the Hoffman boundary we have maximal order of coclique is not any more than l ^ vm/(k + m) = 141. Further, p divides 72 and 765 - l, so either p = 2, l is odd, number x2(g) = (l + 47 - a1(g)/6)/7 is even and a1(g) = 84m + 6l + 282, or p = 3, l divided by 3, number x2(g) = (l + 47 - a1(g)/6)/7 comparable to -1 by modulo 3 and a1 (g) = 126m + 6l + 324.

Let Q contains an edge and Q be an union of isolated cliques. Then p divides 245 and 72, contradiction. □

Lemma 4.2. Let Q contains geodesic 2-path b, a, c. Then

(1) r does not contain its own strongly-regularsubgraphs with parameters (v',k', 42, 72);

(2) p = 5, |Q| = 5l + 4, l < 64, for e G Q we have |Q(e)| = 5t + 3, t = 2,3,..., 9, 210m + 30l + 180;

(3) p = 3, |Q| = 3n, n < 108, for e G Q we have |Q(e)| = 3t, t = 1, 2,..., 8, a1(g) = 18n + 126l + 324;

(4) p = 2, |Q| = 2l+1, l < 161, for e G Q we have |Q(e)| = 2n, n < 12, a1(g) = 12l+84s+288.

Proof. Assume r contains its own strongly-regular subgraph A with parameters (v',k', 42,72). Then n2 = 900 + 4(k' - 72), n = 2u, k' = u2 - 153 and A has nonprincipal eigenvalues u -15, -(u + 15). Further, multiplicity of u - 15 is f = (u + 14)(u2 - 153)(u2 + u - 138)/(144u), contradiction.

By Theorem 1.1 we have p = 2,3,5.

11

5/2 -54/7 47/2 47/7

Let p = 5. Then |Q| = 5l + 4, l < 64 and by theorem 1 subgraph Q(a) contains geodesic 2-path, |Q(a)| = 5t + 3, t = 2, 3,..., 9. Finally, the numder x2(g) = (5l + 51 — a1(g)/6)/7 is comparable to 3 by modulo 5 and a1 (g) = 210m + 30l + 180.

Let p = 3. Then |Q| = 3n, n ^ 108 and by theorem 1 we have |Q(a)| = 3t, t = 1, 2,..., 8, the number x2(g) = (3n + 47 — a1(g)/6)/7 is comparable to -1 by modulo 3 and a1(g) = 18n +126l + 324.

Let p = 2. Then |Q| = 2l +1, l < 161 and by theorem 1 we have |Q(a)| = 2n, n < 12, the number X2 (g) = (2l + 48 — a1 (g)/6)/7 is even and a1 (g) = 12l + 84s + 288. □

Lemmas 4.1, 4.2 imply the Theorem 1.2.

Lemma 4.3. Let the group G = Aut^) acts transitively on the set vertices of Г. Then the following assertions hold:

(1) if f is element of G with order 47, then ^q(f )| divides 94;

(2) S(G) = 1;

(3) if T is socle of group G, then \T| divided by 23.

Proof. Let f be an element of G with order 47, g be an element of Ca (f) with order p < 47. By theorem 2 Fix(f) is empty graph and a1(f) = 282. In view of Theorem 2 and the action of f on Q should be one of the statements:

(1) Q is empty graph, p = 3 and a1(g) = 126l + 72;

(2) Q is l-coclique, p = 2, l is odd, a1 (g) = 84m + 6l + 282 or p = 3, l divided by 3, a1(g) = 126m + 6l + 324;

(3) Q contains geodesic 2-path and either

(i) p = 5, |Q| = 5l + 4, l < 64, a1(g) = 210m + 30l + 180, or

(ii) p = 3, |Q| = 3n, n < 108, a1(g) = 18n + 126l + 324, or

(iii) p = 2, |Q| = 2l + 1, l < 161, a1(g) = 12l + 84s + 288.

If Q is empty graph, then a1(g) = 18(7l + 4) divided by 47, contradiction. If Q is nonempty graph, then |Q| = 47t, p divides 27 — t and either t = 6, p = 3, 7, or t = 5,1, p = 2, or t = 3, p = 2, 3, or t = 2, p = 5. Further, x2(g) = (282t — a1(g))/42 + 47/7, a1(g) = 282w and X2 (g) = 47(t — w + 1)/7. In case p = 5 we have a1(g) = 47 • 30l = 0, t = 2 and x2(g) = 47 • 3/7, contradiction. If Q is coclique, then by Hoffman's boundary we have |Q| < 141, so |Q| = 141 and 4 + w divided by 7. Now, each vertex of Г — Q is adjacent to 36 vertices of Q. This contradicts the fact that for the 3-clique {u, ug ,ug } neighborhood of vertex u in Г is strongly-regular graph with parameters (288,42,6,6).

Let Q contains geodesic 2-path. If p = 3, then |Q| = 3n, n = 47, 94 and a1(g) = 18(n+7l + 18) divided by 47, contradiction. If p = 2, then |Q| = 2l + 1, l = 23, 60,117 and a1(g) = 12(l+7s + 24) divided by 47, so l = 23 and s = 0.

Note, that |Г — Q| = 26 • 47 is not divisible by 4, so CG(f )| is not divisible by 4.

As v = 27 • 47, then by assertion (1) either S(G) = O3(G), or G/O3(G) contains normal subgroup with order 47. Let Q = O3(G) = 1, f be an element of G with order 47. Then

Q : Q^ = 27, and as k = 288, then Qa : Qa^ = 9 for any vertex b £ [a]. Further, the order of Sylow 3-subgroup of symmetric group of degree 42 is 319, and by the theorem 1 the number

|QaьЬ| divides 319. As 323 — 1 divided by 47, and 3* — 1 is not divisible by 47 for i < 23, then Q is elementary Abelian group of order 323. On the other hand, the group Qa,b acts faithfully on [a] П [b] and for c £ [a] П [b] group Qa,b,c has an index that divides 3 in Qa,b, and acts on

([a] П [b] П [c]) U ([a] П [b] — [c]). Hence the subgroup of index 81 from Qa,b,c is embedded in a Sylow 3-subgroup of the symmetric group of degree 27. This contradicts the fact that Qa divided by 318, and order of Sylow 3-subgroup of the symmetric group of degree 27 is 313.

So, 03(G) = 1. By the table 1 from [6] the order of socle T of group G is divided by 23. □

As in wiew of Theorem 2 is not divisible by 23, then G acts intransitive on the set of vertices of graph r. Corollary 1.1 is proved.

5. Automorphisms of graph with intersection array {288, 245,48,1; 1, 24, 245, 288}

Let to the end of paper r be a distance-regular graph with intersection array {288, 245,48,1; 1, 24, 245,288} and with spectrum 288i, 48282, 6io8°, —62256, —36i88, G = Aut(r), g be an element of G with prime order p and Q = Fix(g) .

Lemma 5.1. Let xi be a character of representation 0 on subspace of dimension 282, x4 be a character of representation 0 on subspace of dimension 188. Then

(1) Xi(g) = (12ao(g) + 2ai(g) — a3(g) — 6a4(g))/162, xi(g) — 282 divided by p;

(2) X4(g) = (7ao(g) + alpha2(g) + 7a4(g))/126 — 47/2, x4(g) — 188 divided by p;

(3) if Q is empty graph, then either

(i) p = 3, a4(g) = v, or

(ii) p = 3, a4(g) divided by 9, ai(g) = 126l + 234 + 2a4(g) and a3(g) = 252l — 18 — 2a4(g), l ^ 3, or

(iii) p = 47, a4(g) = 0, alphai(g) = 6 • 47 and alpha3(g) = 12 • 47.

Proof. We have

Q

1 1 1 1 1

282 47 0 -47/2 -141

1080 45/2 -54/7 45/2 1080

2256 -47 0 47/2 -1128

188 -47/2 47/7 -47/2 188

So x1(g) = (12ao(g) + 2a1(g) - a3(g) - 6a4(g))/162, x1(g) - 282 divided by p.

Similarly, x4(g) = (56ao(g) - 7a1(g) + 2a2(g) - 7a3(g) + 56a4(g))/189. As a1 (g) + a3(g) = v - ao(g) - a2(g) - a4(g), then x4(g) = (7ao(g) + alpha2(g) + 7a4(g))/126 - 47/2, x4(g) - 188 divided by p.

Let Q is empty graph. Then p G {3, 47}. If p = 3, then either a4(g) = v, or g induces an automorphism of order 3 of antipodal private r. In last case we have x4(g) = (alpha2(g) + 7a4(g))/126 - 47/2. As a4(g) = 3w4, then a2(g) = 21w2 and x4(g) = (w2 + w4)/6 - 47/2 comparable to -1 by modulo 3. Further, x1 (g) = (2a1(g) - a3(g) - 6a4(g))/162 divided by. In other hand, ao(g) = a4(g)/3 divided by 3 and by theorem 2 we have a1(g) = (a1 (g)+ a3(g))/3 = 126l + 72. So, a1(g) = 126l + 234 + 2a4(g) and a3(g) = 252l - 18 - 2a4(g), l < 3.

If p = 47, then a4(g) = 0, x4(g) = alpha2(g)/126 - 47/2, x1(g) = (2a1(g) - a3(g))/162 are divided by 3. So, alpha2(g) = 63• 47 and alpha1(g) + alpha3(g) = 18-47, and so alpha1(g) = 6• 47, alpha3(g) = 12 • 47.

x4(g) = (a2(g) + 20l -460)/40, the number x4(g) is odd and a2(g) = 20 - 20l - 80s. Further, a1 (g) + a3(g) = 624 + 20l + 80s, the number x1(g) = (a1 (g) - 312 - 18l - 40s)/28 is even, so a1 (g) = 56m + 18l + 40s + 32, a3(g) = 592 + 2l + 40s - 56m.

If p = 7, then a4(g) = 0, x4(g) = a2(g)/40 - 23/2, a2(g) = 280l + 140. Further, a1(g) + a3(g) = 504 - 280l, x1(g) = (a1(g) - 252 + 140l)/28, the number x1(g) comparable to 4 by modulo 7, so a1(g) = 140 - 140l + 196t and a3(g) = 364 - 196t - 140l.

If p = 23, then a4(g) = 0, x4(g) = a2(g)/40 - 23/2, a2(g) = 460. Further, a1 (g) + a3(g) = 184, x1 (g) = (a1(g) - 92)/28, so a1 (g) = a3(g) = 92. □

Lemma 5.2. If element g induces a nontrivial automorphism of the graph r and 0 is non-empty graph, then one of the following assertions hold:

(1) 0 is an union of three isolated n-cliques and either p = 5, n = 4, 9, or p = 7, n = 2, 9;

(2) p = 2, 0 is a n-clique, l-coclique or contains geodesic 2-path;

(3) p = 3, 0 is a l-coclique or contains geodesic 2-path.

Proof. Let g induces a nontrivial automorphism of r. By Theorem 2 either 0 is empty graph, p = 3,47, or 0 is clique or coclique, or 0 contains geodesic 2-path.

If 0 is empty graph, then 0 is emty graph too.

If 0 is n-clique, then by Theorem 2 either p = 2, n =1, ai(g) = 84i + 36, or p = 5, n = 4, ai(g) = 210s + 180 or n = 9, a1(g) = 210s, or p = 7, n = 2, a1(g) = 294t + 42 or n = 9, a1(g) = 294t + 84.

In case p = 2 we have a0(g) + «4(g) = 3, the number x4(g) = (21 + alpha2(g))/126 — 47/2 is even, a2(g) = 126s + 42 and by Theorem 2 we have a2(g) = 1232 — 84i = 42s + 14. Further, a1(g) + a3(g) = 3762 — 126s, the number x1(g) = (6a0(g) + «1(g) — 1274)/54 is even, a1(g) = 108l + 1274 — 6a0(g) and a3(g) = 2498 — 108l + 6a0(g) — 126s.

In casep = 5 we have a4(g) = 0, a0(g) = 3n, the number x4(g) = (21n+alpha2(g))/126—47/2 comparable to 3 by modulo 5, a2(g) = 126t + 63 — 21n. If n = 4, then by Theorem 2 we have a2(g) = 1085 — 210s and x4(g) = (159 — 30s)/6 — 47/2 = 3 — 5s. If n = 9, then by Theorem 2 we have a2(g) = 1260 — 210s and x4(g) = (189 — 30s)/6 — 47/2 = 8 — 5s.

In case p = 7 we have a4(g) = 0, a0(g) = 3n, the number x4(g) = (21n+alpha2(g))/126—47/2 comparable to 6 by modulo 7. If n = 2, then by Theorem 2 we have a2(g) = 1225 — 294t and x4(g) = (59 — 14i — 47)/2. If n = 9, then by Theorem 2 we have a2(g) = 1176 — 294t and x4 (g) = (59 — 14i)/2 — 47/2.

Let 0 be a l-coclique andp = 2. Then l is odd, a2(g) = 987—84m—7l, x4(g) = (141 —12m)/6— 47/2 = —2m. Let 0 be a l-coclique and p = 3, l is divided by 3, a2 (g) = 1269 — 126m — 7l — 324, x4 (g) = (45 — 6m — 47)/2.

Let 0 contains geodesic 2-path. If p = 5, then |01 = 5l + 4, l < 64, a2(g) = 1085 — 210m — 35l, x4(g) = (33 — 10m — 47)/2. If p = 3, then |01 = 3n, n < 108, a2(g) = 945 — 21n — 126l, x4(g) = (45 — 6l — 47)/2. If p =2, then |01 = 2l +1, l < 161, a2(g) = 980 — 14l — 84s, x4 (g) = —2s. □

Lemmas 5.1, 5.2 imply the Theorem 1.3. Theorem 1.3 and Corollary 1.1 imply the Corollary 1.2.

References

[1] A.Jurisic, J.Koolen, Classification of the family AT4(qs, q, q) of antipodal tight graphs, J. Comb. Theory, 118(2011), no. 3, 842-852.

[2] A.E.Brouwer, W.H.Haemers, Spectra of Graphs, New York, Springer, 2012.

[3] P.J.Cameron, Permutation Groups, London Math. Soc. Student Texts, 45(1999).

[4] A.L.Gavrilyuk, A.A.Makhnev, On automorphisms of a distance-regular graph with in-tersetion array {56,45,1; 1, 9, 56}, Doklady Akademii Nauk, 432(2010), no. 5, 512-515 (in Russian).

[5] M.Behbahani, C.Lam, Strongly regular graphs with non-trivial automorphisms, Discrete Math., 311(2011), no. 3, 132-144.

[6] A.V.Zavarnitsine, Finite simple groups with narrow prime spectrum, Siberian Electr. Math. Reports, 6(2009), 1-12.

Автоморфизмы AT4(6,6,3)-графа и отвечающих ему сильно регулярных графов

Константин С. Ефимов

Уральский федеральный университет Мира, 19, Екатеринбург, 620002 Уральский государственный университет экономики 8 марта, 62, Екатеринбург, 620144

Россия

Александр А. Махнев

Институт математики и механики им. Н.Н.Красовского УрО РАН

С.Ковалевской, 4, Екатеринбург, 620990

Уральский федеральный университет Мира, 19, Екатеринбург, 620002

Россия

Кулен и Юришич определили класс АТ4-графов (антиподальных плотных графов диаметра 4). Среди этих графов имеется граф с массивом пересечений {288, 245,48,1; 1, 24, 245, 288} на v = 1 + 288 + 2940 + 576 + 2 = 3807 вершинах. Антиподальное частное этого графа является сильно регулярным графом с параметрами (1269, 288,42, 72). Оба этих графа являются локально псевдо GQ(7, 5)-графами. В работе найдены возможные автоморфизмы указанных графов. В частности, группа автоморфизмов дистанционно регулярного графа с массивом пересечений {288, 245,48,1; 1, 24, 245, 288} действует интранзитивно на множестве его антиподальных классов.

Ключевые слова: дистанционно регулярный граф, сильно регулярный граф, автоморфизм графа.