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

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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

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Amply regular graph We consider undirected graphs without loops or multiple edges. For a vertex a of a graph Γ the subgraph Γi(a) = {b | d(a, b) = i} is called i-neighbourhood of a in Γ. We set [a] = Γ1(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.

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Distance-regular graph

If d(u, w) = i then 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 {b0, b1, ..., bd−1; c1, ..., cd} if bi = bi(u, w) and ci = ci(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 = b0, λ = k − b1 − 1 and µ = c2.

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Partial geometries

A partial geometry pGα(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 zi 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 pGα(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 pGα(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 pGα(s, t).

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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:

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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.

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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:

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Theorem 2

1 ∆ is either a graph with parameters (13, 6, 2, 3) or

(17, 8, 3, 4), or a pseudo-geometric graph for pG2(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}.

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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:

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Proposition 1

1 ∆ is a pseudo-geometric graph for pGt(t + 3, t); 2 ∆ is the complement graph of a pseudo-geometric graph for

pG4(s, 3) (if s = 4u then ∆ is a pseudo-geometric graph for pG3u−3(3u, 4u − 4));

3 m = 1 and ∆ is a union of isolated 4-cliques; 4 ∆ is a conference graph with parameters

(4n + 1, 2n, n − 1, n), n ∈ {7, 9, 10, 11, 12};

5 ∆ belongs to the finite set of exceptional strongly regular

graphs with nonprincipal eigenvalue 3.

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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

(4n + 1, 2n, n − 1, n) and Γ is a Taylor graph with intersection array {4n + 1, 2n, 1; 1, 2n, 4n + 1};

2 [u] is a pseudo-geometric graph for pGt(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 pG4(8, 3) and Γ is a Taylor graph with intersection array {63, 32, 1; 1, 32, 63}.

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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);

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(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);

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(4) (287, 126, 45, 63), (288, 147, 66, 84), (288, 123, 42, 60), (297, 168, 87, 105), (300, 273, 248, 252), (301, 150, 65, 84), (301, 108, 27, 45), (320, 99, 18, 36), (343, 150, 53, 75), (344, 175, 78, 100), (344, 147, 50, 72), (351, 300, 255, 264), (352, 243, 162, 180), (364, 297, 240, 252), (375, 198, 93, 117), (392, 255, 158, 180), (392, 115, 18, 40), (396, 135, 30, 54), (400, 378, 357, 360), (400, 243, 138, 162); (5) (416, 315, 234, 252), (430, 390, 353, 360), (441, 264, 147, 174), (441, 220, 95, 124), (456, 315, 210, 234), (460, 243, 114, 144), (460, 153, 32, 60), (490, 297, 168, 198), (495, 234, 93, 126), (495, 190, 53, 85), (507, 276, 135, 168), (507, 198, 57, 90), (536, 405, 300, 324), (539, 234, 81, 117), (540, 462, 393, 408), (540, 363, 234, 264), (540, 273, 120, 156), (540, 231, 78, 114), (616, 410, 261, 296), (621, 300, 123, 165);

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(6) (625, 378, 213, 252), (640, 243, 66, 108), (651, 468, 327, 360), (672, 451, 290, 328), (676, 315, 122, 168), (689, 432, 255, 297), (690, 318, 121, 168), (690, 273, 80, 126), (726, 300, 95, 144), (729, 408, 207, 255), (735, 318, 109, 159), (736, 675, 618, 630), (736, 555, 410, 444), (736, 371, 162, 212), (736, 315, 106, 156), (768, 531, 354, 396), (784, 435, 218, 270), (800, 423, 198, 252), (845, 588, 395, 441), (850, 513, 288, 342); (7) (891, 570, 345, 399), (896, 675, 498, 540), (936, 561, 312, 372), (976, 675, 450, 504), (980, 801, 648, 684), (1058, 693, 432, 495), (1065, 798, 585, 636), (1081, 486, 177, 252), (1128, 567, 246, 324), (1136, 855, 630, 684), (1156, 819, 562, 624), (1156, 615, 290, 369), (1176, 675, 354, 432), (1197, 780, 483, 555), (1275, 938, 673, 737), (1275, 728, 379, 464), (1288, 1053, 852, 900), (1296, 1110, 945, 984), (1300, 783, 438, 522), (1331, 1026, 777, 837);

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(8) (1331, 850, 513, 595), (1344, 948, 647, 720), (1408, 1323, 1242, 1260), (1470, 1053, 732, 810), (1519, 1242, 1005, 1062), (1536, 1155, 850, 924), (1596, 1485, 1380, 1404), (1625, 1368, 1143, 1197), (1666, 1620, 1575, 1584), (1701, 1380, 1107, 1173), (1711, 1134, 717, 819), (1716, 1323, 1002, 1080), (1716, 1225, 848, 940), (1818, 1518, 1257, 1320), (1825, 1368, 1003, 1092), (1863, 1596, 1359, 1416), (1944, 1407, 990, 1092), (1961, 1890, 1821, 1836), (2080, 1683, 1346, 1428), (2160, 1683, 1290, 1386);

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(9) (2300, 2178, 2061, 2088), (2336, 1755, 1290, 1404), (2527, 1998, 1557, 1665), (3060, 2415, 1878, 2010), (3136, 2565, 2076, 2196), (3186, 2730, 2325, 2424), (3250, 3078, 2913, 2952), (3304, 2835, 2418, 2520), (3381, 2740, 2195, 2329), (3520, 2907, 2378, 2508), (3741, 2970, 2325, 2484), (3872, 3171, 2570, 2718), (4000, 3483, 3018, 3132), (4131, 3540, 3015, 3144), (4992, 4163, 3442, 3620), (5240, 4563, 3954, 4104), (5376, 4875, 4410, 4524), (5500, 4953, 4448, 4572), (5618, 4932, 4311, 4464), (5832, 4998, 4257, 4440);

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(10) (6480, 5673, 4944, 5124), (6776, 6075, 5430, 5589), (7514, 6633, 5832, 6030), (8576, 7875, 7218, 7380), (8625, 7728, 6903, 7107), (9472, 8883, 8322, 8460), (9504, 8385, 7368, 7620), (10051, 9180, 8367, 8568), (10580, 9387, 8298, 8568), (11616, 10695, 9830, 10044), (12750, 12078, 11433, 11592), (14848, 13923, 13042, 13260), (14848, 13635, 12498, 12780), (15456, 14355, 13314, 13572), (16225, 14688, 13263, 13617), (17500, 16578, 15693, 15912), (19965, 18538, 17189, 17524), (23276, 21945, 20672, 20988), (24696, 22935, 21270, 21684), (25025, 23598, 22233, 22572), (27455, 25758, 24141, 24543), (33664, 31563, 29562, 30060), (38875, 36828, 34863, 35352).

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Graphs whose local graphs are exceptional graphs with eigenvalue 3

Theorem 4 [4] Let Γ be an amply regular graph whose local graphs are exceptional graphs with eigenvalue 3. Then parameters of strongly regular graphs do not belong (7 − 10) from the conclusion of Proposition 2. Theorem 5 [14] Let Γ be an amply regular graph whose local graphs are exceptional graphs with eigenvalue 3, u ∈ Γ. If the parameters of [u] belong to (3 − 6) from the conclusion of Proposition 2, then Γ is a strongly regular graph with parameters (490, 297, 168, 198), (616, 287, 126, 140), (640, 243, 66, 73), (961, 320, 99, 110) or (1331, 850, 513, 595).

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Graphs whose local graphs are pseudo-geometric graphs for pGs−3(s, t)

Theorem 6 [10] Let Γ be an amply regular graph whose local graphs are pseudo-geometric graphs for pGs−3(s, t). If d(Γ) ≥ 4, then s = 4,

• r s = 5 and t ≤ 6.

Corollary 1 [10] Let Γ be a distance-regular graph whose local graphs are pseudo-geometric graphs for pGs−3(s, t). If d(Γ) ≥ 4, then one of the following holds:

1 [u] is the 5 × 5-grid and Γ is the Johnson graph J(10, 5); 2 [u] is pseudo-geometric graph for GQ(4, 2) and Γ is the

unique locally GQ(4, 2)-graph with intersection array {45, 32, 12, 1; 1, 6, 32, 45}.

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Theorem 7 [16] Let Γ be an amply regular graph with diameter 3 whose local graphs are pseudo-geometric graphs for pGs−3(s, t). If s > 5, then s = 6 and Γ is a Taylor graph. Corollary 2 [16] Let Γ be a distance-regular graph whose local graphs are pseudo-geometric graphs for pGs−3(s, t). If d(Γ) ≤ 3, s > 4, u ∈ Γ, then either Γ is a strongly regular graph with parameters (117, 36, 15, 9), (232, 81, 30, 27) or (287, 126, 45, 63), or one of the following holds:

1 s = 6 and Γ is a Taylor graph; 2 s = 5, t = 1 and Γ is the halved 7-cube. A.A. Makhnev Graphs whose local graphs are srg with the second eigen

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Extensions of strongly regular graphs with eigenvalue 3

Final Theorem Let Γ be a distance-regular graph, whose local graphs are strongly regular graphs with nonprincipal eigenvalue r, 2 < r ≤ 3. If d(Γ) > 2, u ∈ Γ, then either [u] is a union of isolated 4-cliques, or [u] is a conference-graph with parameters (4n + 1, 2n, n − 1, n) (7 ≤ n ≤ 12), or [u] is a pseudo-geometric graph for pG3(6, t), and Γ is a Taylor graph, or one of the following holds:

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Final Theorem

1 [u] is the 5 × 5-grid and Γ is the Johnson graph J(10, 5), or

[u] is the pseudo-geometric graph for GQ(4, 2) or GQ(4, 6) and Γ has intersection array {45, 32, 12, 1; 1, 6, 32, 45} or {125, 96, 1; 1, 48, 125} respectively;

2 [u] is the triangular graph T(7) and Γ is the halved 7-cube,

• r [u] has parameters (96, 19, 2, 4) and Γ has intersection

array {96, 76, 1; 1, 19, 96};

3 [u] has parameters (115, 18, 1, 3) and Γ has intersection

array {115, 96, 8; 1, 8, 92} or {115, 96, 30, 1; 1, 10, 96, 115},

4 [u] has parameters (99, 14, 1, 2) and Γ has intersection array

{99, 84, 1; 1, 14, 99}, {99, 84, 1; 1, 12, 99} or {99, 84, 30; 1, 6, 54};

5 [u] has parameters (169, 42, 5, 12) or (256, 51, 2, 12) and Γ

has intersection arrays {169, 126, 1; 1, 42, 169} or {256, 204, 1; 1, 51, 256} respectively.

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neighbourhood of each vertex is the complementary graph

• f a Seidel graph, Doklady 2010, v. 434, N 4, 447-449.

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A.A. Makhnev Graphs whose local graphs are srg with the second eigen