Graphs whose local graphs are srg with the second eigenvalue 3 A.A. - PowerPoint PPT Presentation

t1 t2 t3 t4 t4 lit Graphs whose local graphs are srg with the second eigenvalue 3 A.A. Makhnev Institute of Mathematics and Mechanics UB RAS Ekaterinburg, Russia makhnev@imm.uran.ru Villanova, June 2014 A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Amply regular graph We consider undirected graphs without loops or multiple edges. For a vertex a of a graph Γ the subgraph Γ i ( a ) = { b | d ( a, b ) = i } is called i -neighbourhood of a in Γ . We set [ a ] = Γ 1 ( a ) , a ⊥ = { a } ∪ [ a ] . The degree of a vertex a of Γ is the number of vertices in [ a ] . A local graph of Γ is the subgraph induced by [ x ] for a vertex x of Γ . A graph is called regular of degree k , if the degree of any its vertex is equal to k . The graph Γ is called amply regular with parameters ( v, k, λ, µ ) if Γ is regular of degree k on v vertices, and | [ u ] ∩ [ w ] | is equal to λ , if u adjacent to w , and is equal to µ , if d ( u, w ) = 2 . An amply regular graph of diameter 2 is called strongly regular. A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Distance-regular graph If d ( u, w ) = i then by b i ( u, w ) (by c i ( u, w ) ) we denote the number of vertices in Γ i +1 ( u ) ∩ [ b ] (in Γ i − 1 ( u ) ∩ [ b ] ). The graph Γ with diameter d is called distance-regular with intersection array { b 0 , b 1 , ..., b d − 1 ; c 1 , ..., c d } if b i = b i ( u, w ) and c i = c i ( u, w ) for every i ∈ { 0 , ..., d } and for every vertices u, w with d ( u, w ) = i . Distance-regular with diameter 2 is called strongly regular with parameters ( v, k, λ, µ ) , where v is the number of vertices of the graph, k = b 0 , λ = k − b 1 − 1 and µ = c 2 . A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Partial geometries A partial geometry pG α ( s, t ) is a geometry of points and lines such that every line has exactly s + 1 points, every point is on t + 1 lines (with s > 0 , t > 0 ) and for any antiflag ( P, y ) there are exactly α lines z i containing P and intersecting y . In the case α = 1 we have generalized quadrangle GQ ( s, t ) . Pseudo-geometric graph The point graph of a partial geometry pG α ( s, t ) has points as vertices and two points are adjacent if they are incident to the same line. The point graph of a partial geometry pG α ( s, t ) is strongly regular with parameters v = ( s + 1)(1 + st/α ) , k = s ( t + 1) , λ = s − 1 + ( α − 1) t , µ = α ( t + 1) . A strongly regular graph with these parameters for some natural numbers s, t, α is called pseudo-geometric graph for pG α ( s, t ) . A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Graphs whose local graphs are srg with eigenvalue 1 Recently, the program of investigations of distance-regular graphs whose local graphs are strongly regular graphs with the second eigenvalue at most 1 was completed [1]. Theorem 1. Let Γ be a distance-regular graph, whose local graphs are strongly regular graphs with the second eigenvalue at most 1, let u be a vertex of Γ and ∆ = [ u ] . Then ∆ is a union of isolated edges or one of the following holds: A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Theorem 1 1 d (Γ) = 2 , ∆ is the complement of an n × n -grid (or of Shrikhande graph) and Γ is the complement of ( n + 1) × ( n + 1) -grid, or ∆ is the complement of a triangular graph T ( n ) (or of one of Chang graphs) and Γ is the complement of T ( n + 2) or n = 6 and Γ has parameters (36 , 15 , 6 , 6) or (28 , 15 , 6 , 10) , or ∆ is the complement of Petersen graph, Clebsch graph or Schlafli graph and Γ is Clebsch graph (locally T (5) ), or Γ is locally GQ(2,4)-graph with parameters (64 , 27 , 10 , 12) ; 2 d (Γ) > 2 , either ∆ is the pentagon or a pseudo-geometric graph for GQ (2 , t ) , t ∈ { 1 , 2 , 4 } and Γ is a Taylor graph (in particular, ∆ is the 3 × 3 -grid and Γ is the Johnson graph J (6 , 3) ), or ∆ is the complement of Clebsch graph and Γ has intersection array { 16 , 10 , 1; 1 , 5 , 16 } , or ∆ is the Petersen graph and Γ is the Conway-Smith graph or the Doro graph. A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Graphs whose local graphs are srg with eigenvalue 2 Recently, the program of investigations of distance-regular graphs whose local graphs are strongly regular graphs with the second eigenvalue r , 1 < r ≤ 2 was completed [2]. Theorem 2. Let Γ be a distance-regular graph, whose local graphs are strongly regular graphs with the second eigenvalue 2, u is a vertex of Γ and ∆ = [ u ] . Then ∆ is a union of isolated triangles or one of the following holds: A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Theorem 2 1 ∆ is either a graph with parameters (13 , 6 , 2 , 3) or (17 , 8 , 3 , 4) , or a pseudo-geometric graph for pG 2 (4 , t ) , t ∈ { 1 , 2 , 3 , 7 , 9 , 12 , 17 , 27 } and Γ is a Taylor graph; 2 ∆ is the 4 × 4 -grid and Γ is the Johnson graph J (8 , 4) ; 3 ∆ is the Hoffman-Singleton graph and Γ is a Terwilliger graph with intersection array { 50 , 42 , 1; 1 , 2 , 50 } or { 50 , 42 , 9; 1 , 2 , 42 } ; 4 ∆ is a Gewirtz graph and Γ has intersection array { 56 , 45 , 16 , 1; 1 , 8 , 45 , 56 } ; 5 ∆ has parameters (81 , 20 , 1 , 6) and Γ has intersection array { 81 , 60 , 1; 1 , 20 , 81 } . A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Graphs whose local graphs are srg with eigenvalue 3 Initial reduction A.A. Makhnev suggested the program of investigations of amply regular graphs whose local graphs are strongly regular graphs with the second eigenvalue 3. Proposition 1 [3]. Let ∆ be a strongly regular graph with nonprincipal eigenvalues r, − m , 2 < r ≤ 3 . Then one of the following holds: A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Proposition 1 1 ∆ is a pseudo-geometric graph for pG t ( t + 3 , t ) ; 2 ∆ is the complement graph of a pseudo-geometric graph for pG 4 ( s, 3) (if s = 4 u then ∆ is a pseudo-geometric graph for pG 3 u − 3 (3 u, 4 u − 4) ); 3 m = 1 and ∆ is a union of isolated 4 -cliques; 4 ∆ is a conference graph with parameters (4 n + 1 , 2 n, n − 1 , n ) , n ∈ { 7 , 9 , 10 , 11 , 12 } ; 5 ∆ belongs to the finite set of exceptional strongly regular graphs with nonprincipal eigenvalue 3. A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Reduction to locally exceptional graphs Theorem 3 [3]. Let Γ be a distance-regular whose local graphs are strongly regular graphs with the second eigenvalue 3 and let u be a vertex of Γ . Then either [ u ] is a union of isolated 4 -cliques, or [ u ] is an exceptional graph, or one of the following holds: 1 [ u ] is a conference graph with parameters (4 n + 1 , 2 n, n − 1 , n ) and Γ is a Taylor graph with intersection array { 4 n + 1 , 2 n, 1; 1 , 2 n, 4 n + 1 } ; 2 [ u ] is a pseudo-geometric graph for pG t ( t + 3 , t ) and either ( i ) t = 1 , [ u ] is the 5 × 5 -grid and Γ is the Johnson graph J (10 , 5) , or ( ii ) t = 3 , [ u ] has parameters (49 , 24 , 11 , 12) and Γ is a Taylor graph with intersection array { 49 , 24 , 1; 1 , 24 , 49 } ; 3 [ u ] is the complement graph of a pseudo-geometric graph for pG 4 (8 , 3) and Γ is a Taylor graph with intersection array { 63 , 32 , 1; 1 , 32 , 63 } . A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit Exceptional graphs with eigenvalue 3 and their extensions Exceptional graphs In [4], A.A. Makhnev and D.V. Paduchikh have obtained parameters of exceptional graphs with eigenvalue 3. Proposition 2. Let ∆ be an exceptional graph with nonprincipal eigenvalue 3. Then Γ has parameters from one of the following lists: (1) graphs without triangles: (162 , 21 , 0 , 3) , (176 , 25 , 0 , 4) , (210 , 33 , 0 , 6) , (266 , 45 , 0 , 9) ; graphs with λ = 1 : (99 , 14 , 1 , 2) , (115 , 18 , 1 , 3) ; graphs with λ = 2 : (96 , 19 , 2 , 4) , (196 , 39 , 2 , 9) ; graphs with λ = 3 : (45 , 12 , 3 , 3) , (85 , 20 , 3 , 5) , (125 , 28 , 3 , 7) , (165 , 36 , 3 , 9) , (225 , 48 , 3 , 12) , (245 , 52 , 3 , 13) , (325 , 68 , 3 , 17) ; graphs with 3 < λ ≤ 6 : (196 , 45 , 4 , 12) , (21 , 10 , 5 , 4) , (111 , 30 , 5 , 9) , (169 , 42 , 5 , 12) , (88 , 27 , 6 , 9) , (144 , 39 , 6 , 12) ; A.A. Makhnev Graphs whose local graphs are srg with the second eigen

t1 t2 t3 t4 t4 lit (2) (35 , 18 , 9 , 9) , (36 , 21 , 12 , 12) , (40 , 27 , 18 , 18) , (50 , 28 , 15 , 16) , (56 , 45 , 36 , 36) , (64 , 27 , 10 , 12) , (81 , 30 , 9 , 12) , (85 , 54 , 33 , 36) , (96 , 75 , 58 , 60) , (96 , 35 , 10 , 14) , (99 , 84 , 71 , 72) , (100 , 33 , 8 , 12) , (119 , 54 , 21 , 27) , (120 , 63 , 30 , 36) , (120 , 51 , 18 , 24) , (121 , 36 , 7 , 12) , (126 , 45 , 12 , 18) , (133 , 108 , 87 , 90) , (136 , 105 , 80 , 84) , (147 , 66 , 25 , 33) , (148 , 77 , 36 , 44) , (148 , 63 , 22 , 30) ; (3) (162 , 138 , 117 , 120) , (171 , 60 , 15 , 24) , (175 , 108 , 63 , 72) , (176 , 135 , 102 , 108) , (205 , 108 , 51 , 63) , (205 , 68 , 15 , 26) , (208 , 81 , 24 , 36) , (216 , 129 , 72 , 84) , (216 , 75 , 18 , 30) , (220 , 135 , 78 , 90) , (236 , 180 , 135 , 144) , (243 , 66 , 9 , 21) , (245 , 108 , 39 , 54) , (246 , 126 , 57 , 72) , (246 , 105 , 36 , 51) , (246 , 85 , 20 , 34) , (250 , 153 , 88 , 102) , (276 , 165 , 92 , 108) , (276 , 75 , 10 , 24) , (280 , 243 , 210 , 216) ; A.A. Makhnev Graphs whose local graphs are srg with the second eigen