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Symmetric Graphs and Their Automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Symmetric graphs and their automorphisms A.A. Makhnev Institute of Mathematics and Mechanics UB RAS Ekaterinburg, Russia [email protected] Berlin, September 2009 A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Amply regular graph We consider undirected graphs without loops or multiple edges. Let Γ be a graph. For vertex a of Γ the subgraph Γi(a) = fb jd(a; b) = ig is called i-neighboorhod of a in Γ. We ? set [a] = Γ1(a), a = fag [ [a]. The degree of a vertex a of Γ is the number of vertices in [a]. Γ is called regular of degree k, if the degree of any its vertex is equal k. Γ is called amply regular with parameters (v; k; λ, µ) if Γ is regular of degree k on v vertices, and j[u] \ [w]j is equal λ, if u is adjacent to w, is equal µ, if d(u; w) = 2. Amply regular graph of diameter 2 is called strongly regular. A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Some strongly regular graphs Let X be a set of power m. Triangular graph T (m) has the X vertex set 2 = ffu; wg j u; w 2 X; u 6= wg and fu; wg is adjacent to fx; yg, if and only if jfu; wg \ fx; ygj = 1. Let Y be a set of power n. Grid graph m × n has the vertex set X × Y and (x1; y1) is adjacent to (x2; y2), if and only if x1 = x2 or y1 = y2. A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Distance-regular graph Let u; w 2 Γ such that d(u; w) = i. By bi(u; w) (by ci(u; w)) we denote the number of vertices in Γi+1(u) \ [b] (in Γi−1(u) \ [b]). The graph Γ with diameter d is called distance-regular with intersection array fb0; b1; :::; bd−1; c1; :::; cdg if for every i 2 f0; :::; dg the values bi = bi(u; w) and ci = ci(u; w) do not depend on the choice of vertices u; w at distance i.A distance-regular graph with diameter 2 is strongly regular with parameters (v; k; λ, µ), where v is the number of vertices of the graph, k = b0, λ = k − b1 − 1 and µ = c2. A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Higman method Automorphism Let g be an automorphism of a graph Γ. Denote by αi(g) the number of vertices u such that d(u; ug) = i and by Fix(g) the subgraph fa 2 Γ j ag = ag. Intersection numbers Let Γ be a distance-regular graph of diameter d on v vertices. Let us consider the symmetric association scheme (X; R) with d classes, where X is the set of vertices of Γ and 2 Ri = f(u; w) 2 X j d(u; w) = ig. For vertex u 2 X set ki = jΓi(u)j. Let Ai be the adjacency matrix of the graph Γi, P l wich is corresponding to the relation Ri. Then AiAj = pijAl l for some integer numbers pij ≥ 0, which are called the l intersection numbers. Note that pij = jΓi(u) \ Γj(w)j for any vertices u; w with d(u; w) = l. A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Eigenmatrices l Let Pi be the matrix in which in the (j; l) entry there is pij. Then the eigenvalues k = p1(0); :::; p1(d) of the matrix P1 are eigenvalues of Γ with multiplicities m0 = 1; :::; md. Note that the matrix Pi is the value of some integer polynom 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 Γ and PQ = QP = vI holds, where I is the identity matrix of order d + 1. Proposition 1 [1, Theorem 17.12] 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=huj; wji. A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Character In fact, from the proof of the Theorem 17.12 we have that wj are the columns of the matrix P and mjuj are the rows of the matrix Q. The permutation representation of the group G ≤ Aut(Γ) on the vertex set of Γ naturally gives the matrix representation of G v in GL(v; C). The space C is the orthogonal direct sum of the eigenspaces W0;W1; :::; Wd of the adjacency matrix A1 of Γ. For every g 2 G we have (g)A1 = A1 (g), so the subspace Wi is (G)-invariant. Let χi be a character of the representation Wi . Then for g 2 G we obtain [2,§3.7] −1 Pd χi(g) = v j=0 Qijαj(g), where αj(g) is the numbers of vertices x of X such that d(x; xg) = j. A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Note that the value of character is an integer algebraic number, and if the numbers Qij are rational then χi(g) is integer. The Higman method was published in [2]. And in [2] this method was applied only to involutive automorphisms of strongly regular graph with parameters (3250,57,0,1). A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Automorphisms of srg If g is an automorphism of a strongly regular graph Γ, then either Γ has parameters k = 2µ and λ = µ − 1 (a conference graph), or χi(g) is integer. P. Cameron noted that the main difficulte in the Higman method is the calculation of parameters αj(g). But in the class of graphs without triangles, for every automorphism f of ofder 3 we obtaine α1(f) = 0. Apart from, the structure of subgraphs of fixed points of automorphism is strongly restricted in graphs with small λ and µ. For example, M. Aschbacher noted that nonempty subgraph of fixed points of automorphism of strongly regular Moore graph (graph with λ = 0 and µ = 1) is Moore graph or star. The common properties of strongly regular graphs are in A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Lemma 1 Let Γ be a strongly regular graph with parameters (v; k; λ, µ). Then v − k − 1 = k(k − λ − 1)/µ and one of the following holds: 1 k = 2µ, λ = µ − 1 and v = 4µ + 1 is the sum of two squares of some integers; 2 (λ − µ)2 + 4(k − µ) is the square of some positive integer n, and Γ has spectrum k1; rf ; sv−f−1, where r = (λ − µ + n)=2, s = (λ − µ − n)=2 and f = (s + 1)k(s − k)=(nµ). J. Seidel poset the following Seidel Problem Does a strongly regular graph with parameters (99; 14; 1; 2) exist? A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Let Γ be a strongly regular graph with parameters (99; 14; 1; 2), g be an automorphism of Γ. Then for every vertex a 2 Γ the subgraph [a] is the union of 7 isolated edges, Γ has spectrum 141; 354; −444, and 0 1 1 1 1 Q = @ 44 −88=7 11=7 A : 54 81=7 −18=7 A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit So the value of character, obtaining by projection of monomial representation on the eigenspace W2 of the dimension 54 is equal 1 81 18 χ (g) = (54α (g) + α (g) − α (g) = 2 99 0 7 1 7 2 1 (42α (g) + 9α (g) − 2α (g): 77 0 1 2 By substitution α2(g) = 99 − α0(g) − α1(g) we have χ2(g) = (4α0(g) + α1(g) − 18)=7: A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Automorphisms of Seidel graph Involutive automorphisms A. Makhnev and I. Minakova investigated involutive automorphisms of strongly regular graph with parameters (99; 14; 1; 2). Let t be an involutive automorphisms of strongly regular graph with parameters (v; k; 1; 2), Ω = Fix(t). If a 2 Ω; b 2 Ω − a?, then [a] \ [b] = fu; utg is a 2-coclique. If u 2 [a] − Ω and fu; utg is a 2-coclique, then [u] \ [ut] = fa; bg and b 2 Ω − a?. So jΩj = 1 + jΩ(a)j + 2l, where l is the number of t-orbits of size 2 on the edges in [a]. A.A. Makhnev Graphs and automorphisms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 11 lit Proposition 2 Let Γ be a strongly regular graph with parameters (99; 14; 1; 2), having involutive automorphism t.
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