A Q-polynomial Shilla graph with b=5 has intersection arrays {105t,4(21t+1),16(t+1);1,4(t+1),84t}, t∈{3,4,19}. The paper proves that distance-regular graphs with these intersection arrays do not exist. Moreover, feasible intersection arrays of Q-polynomial Shilla graphs with b=6 are found.This work was supported by RFBR and NSFC (project № 20-51-53013)

Belousov, I. N.

Makhnev, A. A.

Ural mathematical journal

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

URAL MATHEMATICAL JOURNAL, Vol. 7, No. 2, 2021, pp. 51–58DOI: 10.15826/umj.2021.2.004SHILLA GRAPHS WITH b = 5 AND b = 61Alexander A. Makhnev†, Ivan N. Belousov††Krasovskii Institute of Mathematics and Mechanics,Ural Branch of the Russian Academy of Sciences,16 S. Kovalevskaya Str., Ekaterinburg, 620990, RussiaUral Federal University,19 Mira str., Ekaterinburg, 620002, Russia†makhnev@imm.uran.ru, ††i belousov@mail.ruAbstract: A Q-polynomial Shilla graph with b = 5 has intersection arrays {105t, 4(21t + 1), 16(t + 1);1, 4(t+ 1), 84t}, t ∈ {3, 4, 19}. The paper proves that distance-regular graphs with these intersection arraysdo not exist. Moreover, feasible intersection arrays of Q-polynomial Shilla graphs with b = 6 are found.Keywords: Shilla graph, Distance-regular graph, Q-polynomial graph.1. IntroductionWe consider undirected graphs without loops or multiple edges. For a vertex a of a graph Γ,denote by Γi(a) the ith neighborhood of a, i.e., the subgraph induced by Γ on the set of all verticesat distance i from a. Define [a] = Γ1(a) and a⊥ = {a} ∪ [a].Let Γ be a graph, and let a, b ∈ Γ. Denote by µ(a, b) (by λ(a, b)) the number of vertices in[a] ∩ [b] if a and b are at distance 2 (are adjacent) in Γ. Further, the induced [a] ∩ [b] subgraph iscalled µ-subgraph (λ-subgraph).If vertices u and w are at distance i in Γ, then we denote by bi(u,w) (by ci(u,w)) the number ofvertices in the intersection of Γi+1(u) (of Γi−1(u), respectively) with [w]. A graph Γ of diameter d iscalled distance-regular with intersection array {b0, b1, . . . , bd−1; c1, . . . , cd} if, for each i = 0, . . . , d,the values bi(u,w) and ci(u,w) are independent of the choice of vertices u and w at distance iin Γ. Define ai = k − bi − ci. Note that, for a distance regular graph, b0 is the degree of thegraph and a1 is the degree of the local subgraph (the neighborhood of the vertex). Further, forvertices x and y at distance l in the graph Γ, denote by plij(x, y) the number of vertices in thesubgraph Γi(x) ∩ Γj(y). The numbers plij(x, y) are called the intersection numbers of Γ (see [2]).In a distance-regular graph, they are independent of the choice of x and y.A Shilla graph is a distance-regular graph Γ of diameter 3 with second eigenvalue θ1 equal toa = a3. In this case, a divides k and b is defined by b = b(Γ) = k/a. Morover, a1 = a− b and Γ hasintersection array {ab, (a+1)(b− 1), b2 ; 1, c2, a(b− 1)}. Feasible intersection arrays of Shilla graphsare found in [6] for b ∈ {2, 3}.Feasible intersection arrays of Shilla graphs are found in [1] for b = 4 (50 arrays) and for b = 5(82 arrays). At present, a list of feasible intersection arrays of Shilla graphs for b = 6 is unknown.Moreover, the existence of Q-polynomial Shilla graphs with b = 5 also is unknown.In this paper, we find feasible intersection arrays of Q-polynomial Shilla graphs with b = 6 andprove that Q-polynomial Shilla graphs with b = 5 do not exist.1This work was supported by RFBR and NSFC (project № 20-51-53013).52 Alexander A. Makhnev and Ivan N. BelousovTheorem 1. A Q-polynomial Shilla graph with b = 6 has intersection array(1) {42t, 5(7t + 1), 3(t + 3); 1, 3(t + 3), 35t}, where t ∈ {7, 12, 17, 27, 57};(2) {372, 315, 75; 1, 15, 310}, {744, 625, 125; 1, 25, 620} or {930, 780, 150; 1, 30, 775};(3) {312, 265, 48; 1, 24, 260}, {624, 525, 80; 1, 40, 520}, {1794, 1500, 200; 1, 100, 1495} or{5694, 4750, 600; 1, 300, 4745}.In view of Theorem 2 from [1], a Q-polynomial Shilla graph with b = 5 has intersection array{105t, 4(21t + 1), 16(t + 1); 1, 4(t + 1), 84t}, t ∈ {3, 4, 19}.Theorem 2. Distance-regular graphs with intersection arrays {315, 256, 64; 1, 16, 252} and{1995, 1600, 320; 1, 80, 1596} do not exist.Theorem 3. Distance-regular graphs with intersection array {420, 340, 80; 1, 20, 336} do notexist.2. Proof of Theorem 1In this section, Γ is a Q-polynomial Shilla graph with b = 6. Then (a2 − 5a− 6)2 − 4(5b2 − a2)is the square of an integer. By [6, Lemma 8], we have2a ≤ c2b(b+ 1) + b2 − b− 2;therefore, a ≤ 21c2+14. It follows from the proof of Theorem 9 in [6] that either k < b3− b = 6 ·35or v < k(2b3 − b + 1) = 428k. By [6, Corollary 17 and Theorem 20], the number b2 + c2 dividesb(b− 1)b2 and−34 = −b2 + 2 ≤ θ3 ≤ −b2(b+ 3)/(3b + 1) ≤ −18.Theorem 2 from [7] implies the following lemma.Lemma 1. If b2 = c2, then Γ has an intersection arrays {42t, 5(7t+1), 3(t+3); 1, 3(t+3), 35t}and t ∈ {7, 12, 17, 27, 57}.To the end of this section, assume that b2 6= c2 and k > θ1 > θ2 > θ3 are eigenvalues of thegraph Γ. Then6(6b2 + c2)/(b2 + c2) = −θ3.On the other hand, according to [6, Lemma 10], the number c2 divides (a + 6)b2, 30a(a + 1) and(a+ 6)b2 ≥ (a+ 1)c2.Lemma 2. If −34 ≤ θ3 ≤ −18, then one of the following statements holds:(1) θ3 = −31 and Γ has one of the intersection arrays {372, 315, 75; 1, 15, 310},{744, 625, 125; 1, 25, 620}, and {930, 780, 150; 1, 30, 775};(2) θ3 = −26 and Γ has one of the intersection arrays {312, 265, 48; 1, 24, 260},{624, 525, 80; 1, 40, 520}, {1794, 1500, 200; 1, 100, 1495}, and {5694, 4750, 600; 1, 300, 4745};(3) θ3 = −21 and Γ has one of the intersection arrays {42t, 5(7t+1), 3(t+3); 1, 3(t+3), 35t} fort ∈ {7, 12, 17, 27, 57}.Shilla graphs with b = 5 and b = 6 53P r o o f. By [6, Lemma 10], c2 divides b(b− 1)b2 = 30b2 and, by [6, Corollary 17], the smallestnonprinciple eigenvalue θ3 is equal to b(bb2 + c2)/(b2 + c2). Therefore, 30(θ3 + 6)/(θ3 + 36) is aninteger and θ3 ∈ {−34,−33,−32,−31,−30,−27,−26,−24,−21,−18}.Let θ3 = −34. Then 3(6b2 + c2) = 17(b2 + c2) and b2 = 14c2. Further, θ3 is a root of theequation x2 − (a1 + a2 − k)x+ (b− 1)b2 − a2 = 0; therefore, a = 425/28 · c2 − 34. In this case, themultiplicity of the first nonprincipal eigenvalue is m1 = 6/5 · (2545c2 − 5544)/c2 , a contradictionwith the fact that 5 does not divide 6 · 5544.Let θ3 = −33. Then 2(6b2 + c2) = 11(b2 + c2) and b2 = 9c2. Further, a = 275/27 · c2 − 33 andthe multiplicity of the first nonprincipal eigenvalue is equal to m1 = 6/5 · (1645c2 − 5184)/c2 , acontradiction as above.Let θ3 = −32. Then 3(6b2 + c2) = 16(b2 + c2) and 2b2 = 13c2. Further, a = 100/13 · c2− 32 andthe multiplicity of the first nonprincipal eigenvalue is m1 = 6/5 ·(1195c2−4836)/c2, a contradictionas above.Let θ3 = −31. Then 6(6b2 + c2) = 31(b2 + c2) and b2 = 5c2. Further, a = 31/5 · c2 − 31and the multiplicity of the first nonprincipal eigenvalue is m1 = 30(37c2 − 180)/c2 = 1110 −5400/c2. The number of vertices in the graph is 31/5 · (222c22 −2005c2 +4500)/c2 ; hence, c2 divides900 and is a multiple of 5. By computer enumeration, we find that, only for c2 = 15, 25 and30, we have admissible intersection arrays {372, 315, 75; 1, 15, 310}, {744, 625, 125; 1, 25, 620} and{930, 780, 150; 1, 30, 775}.Let θ3 = −30. Then (6b2 + c2) = 5(b2 + c2) and b2 = 4c2. Further, a = 125/24 · c2 − 30 and themultiplicity of the first nonprincipal eigenvalue is m1 = 6/5 · (745c2 − 4176)/c2 , a contradiction asabove.Let θ3 = −27. Then 2(6b2 + c2) = 9(b2 + c2) and 3b2 = 7c2. Further, a = 25/7 · c2 − 25 and themultiplicity of the first nonprincipal eigenvalue is m1 = 6/5 · (445c2 − 3276)/c2 , a contradiction asabove.Let θ3 = −26. Then 3(6b2 + c2) = 13(b2 + c2) and b2 = 2c2. Further, a = 13/4 · c2 − 26 andthe multiplicity of the first nonprincipal eigenvalue is m1 = 6(77c2 − 600)/c2 = 462 − 3600/c2.The number of vertices in the graph is 13/8 · (231c22 − 3340c2 + 12000)/c2 ; hence, c2 divides1200 and is a multiple of 4. By computer enumeration, we find that only for c2 = 24, 40, 100,and 300 we have admissible intersection arrays {312, 265, 48; 1, 24, 260}, {624, 525, 80; 1, 40, 520},{1794, 1500, 200; 1, 100, 1495}, and {5694, 4750, 600; 1, 300, 4745}.Let θ3 = −21. Then 2(6b2 + c2) = 7(b2 + c2) and b2 = c2. Further, a = 7/3 · c2 − 21 and themultiplicity of the first nonprincipal eigenvalue is m1 = 6(41c2 − 360)/c2 = 246 − 2160/c2. Thenumber of vertices in the graph is 7/3 · (82c22 − 1335c2 + 5400)/c2 ; hence, c2 divides 1080 and is amultiple of 3. By computer enumeration, we find that, only for c2 = 18, 30, 45, 60, 90, and 180, wehave admissible intersection arrays {42t, 5(7t+1), 3(t+3); 1, 3(t+3), 35t} for t ∈ {3, 7, 12, 17, 27, 57}.A graph with the array obtained for t = 3 does not exist by [5].Let θ3 = −18. Then 6(6b2 + c2) = 19(b2 + c2), so 3b2 = 2c2. Further, a = 2512 · c2 − 18 and themultiplicity of the first nonprincipal eigenvalue is m1 = 6/5 · (145c2 − 1224)/c2, a contradiction.The lemma is proved. Theorem 1 follows from Lemmas 1–2.3. Triple intersection numbersIn the proof of Theorem 3, the triple intersection numbers [3] are used.54 Alexander A. Makhnev and Ivan N. BelousovLet Γ be a distance-regular graph of diameter d. If u1, u2, u3 are vertices of the graph Γ, thenr1, r2, r3 are non-negative integers not greater than d. Denote by{u1u2u3r1r2r3}the set of verticesw ∈ Γ such that d(w, ui) = ri and by[u1u2u3r1r2r3]the number of vertices in{u1u2u3r1r2r3}. The numbers[u1u2u3r1r2r3]are called the triple intersection numbers. For a fixed triple of vertices u1, u2, u3, insteadof[u1u2u3r1r2r3], we will write [r1r2r3]. Unfortunately, there are no general formulas for the numbers[r1r2r3]. However, [3] outlines a method for calculating some numbers [r1r2r3].Let u, v, w be vertices of the graph Γ, W = d(u, v), U = d(v,w), and let V = d(u,w). Sincethere is exactly one vertex x = u such that d(x, u) = 0, then the number [0jh] is 0 or 1. Hence[0jh] = δjW δhV . Similarly, [i0h] = δiW δhU and [ij0] = δiUδjV .Another set of equations can be obtained by fixing the distance between two vertices from{u, v, w} and counting the number of vertices located at all possible distances from the third:d∑l[ljh] = pUjh − [0jh]d∑l[ilh] = pVih − [i0h]d∑l[ijl] = pWij − [ij0](3.1)However, some triplets disappear. For |i − j| > W or i + j < W , we have pWij = 0; therefore,[ijh] = 0 for all h ∈ {0, ..., d}.We setSijh(u, v, w) =d∑r,s,t=0QriQsjQth[uvwrst].If the Krein parameter qhij = 0, then Sijh(u, v, w) = 0.We fix vertices u, v, w of a distance-regular graph Γ of diameter 3 and set{ijh} ={uvwijh}, [ijh] =[uvwijh], [ijh]′ =[uwvihj], [ijh]∗ =[vuwjih], [ijh]∼ =[wvuhji].Calculating the numbers[ijh]′ =[uwvihj], [ijh]∗ =[vuwjih], [ijh]∼ =[wvuhji](symmetrization of the triple intersection numbers) can give new relations that make it possible toprove the nonexistence of a graph.4. Graphs with intersection arrays {315, 256, 64; 1, 16, 252} and{1995, 1600, 320; 1, 80, 1596}Let Γ be a distance-regular graph with intersection array {315, 256, 64; 1, 16, 252}. By [2, The-orem 4.4.3], the eigenvalues of the local subgraph of the graph Γ are contained in the interval[−5, 59/5). Since the Terwilliger polynomial (see [4]) is −4(5x − 59)(x + 5)(x + 1)(x − 43), thenthese eigenvalues lie in [−5,−1]∪(59/5.43]. Hence, all nonprinciple eigenvalues are negative and theShilla graphs with b = 5 and b = 6 55local subgraph is a union of isolated (a1 +1)-cliques, a contradiction with the fact that a1+1 = 49does not divide k = 315.Thus, a distance-regular graph with intersection array {315, 256, 64; 1, 16, 252} does not exist.Let Γ be a distance-regular graph with intersection array {1995, 1600, 320; 1, 80, 1596}. ThenΓ has 1 + 1995 + 39900 + 8000 = 49896 vertices, spectrum 19951, 399495, 1523275,−2126125, and thedual matrix of eigenvaluesQ =1 495 23275 261251 99 175 −2751 0 −56 551 −99/4 931/4 −209.The Terwilliger polynomial of the graph Γ is −20(x + 5)(x + 1)(x − 79)(x − 299); hence, theeigenvalues of the local subgraph are contained in [−5,−1] ∪ {79} ∪ {394}.Note that the multiplicity m1 = 495 of the eigenvalue θ1 = 399 is less than k. By the corollaryto Theorem 4.4.4 from [2] for b = b1/(θ1+1) = 4, the graph Σ = [u] has an eigenvalue −1− b = −5of multiplicity at least k −m1 = 1500.Let the number of eigenvalues 79 of the graph Σ be equal to y. Then the sum of eigenvalues ofthe graph Σ is at most −7500 − (494 − y) + 79y + 394; therefore, y ≥ 95. Now twice the numberof edges in Σ is equal to786030 = 1995 · 394 =∑imiθ2ibut not less than25 · 1500 + 399 + 95 · 792 + 3942 = 786030.Hence, Σ has spectrum 3941.7995,−1399,−51500.Now the number t = kΣλΣ/2 of triangles in Σ containing this vertex is equal to∑i miθ3i /(2v).Therefore,t =∑imiθ3i /(2v) = (3943 + 793 · 95− 399 − 125 · 1500)/3990 = 27021and λΣ = 54042/394 is approximately equal to 137.16, a contradiction.Thus, a distance-regular graph with intersection array {1995, 1600, 320; 1, 80, 1596} does notexist.Theorem 2 is proved.5. Graph with array {420, 340, 80; 1, 20, 336}Let Γ be a distance-regular graph with intersection array {420, 340, 80; 1, 20, 336}. Then Γis a formally self-dual graph having 1 + 420 + 7140 + 1700 = 9261 vertices, spectrum4201, 84420, 07140,−211700, and the dual matrix of eigenvaluesQ =1 420 7140 17001 84 0 −851 0 −21 201 −21 84 −64.The Terwilliger polynomial of the graph Γ is −20(x+5)(x+1)(x−16)(x−59) and the eigenvaluesof the local subgraph are contained in [−5,−1] ∪ {16} ∪ {79}. If the nonprinciple eigenvalues of alocal subgraph are negative, then this subgraph is a union of isolated (a1+1)-cliques, a contradictionwith the fact that a1+1 = 80 does not divide k = 420. Hence, the local subgraph has eigenvalue 6.56 Alexander A. Makhnev and Ivan N. BelousovLemma 3. Intersection numbers of a graph Γ satisfy the equalities(1) p111 = 79, p121 = 340, p132 = 1360, p122 = 5440, p133 = 340,(2) p211 = 20, p212 = 320, p213 = 80, p222 = 5519, p223 = 1300, p233 = 320;(3) p312 = 336, p313 = 84, p322 = 5460, p323 = 1344, p333 = 271.P r o o f. Direct calculations. Let u, v, and w be vertices of a graph Γ, [rst] =[uvwrst], Ω = Γ3(u), and let Σ = Ω2. Then Σ is aregular graph of degree 1344 on 1700 vertices.Lemma 4. Let d(u, v) = d(u,w) = 3 and d(v,w) = 1. Then the following equalities hold :(1) [122] = 2r6/5− 136, [123] = [132] = −2r6/5 + 472, [133] = 2r6/5− 388;(2) [211] = r6/10 − 38, [212] = [221] = −r6/10 + 374, [222] = −14r6/10 + 5576,[223] = [232] = 3r6/2− 490, [233] = −3r6/2 + 1834;(3) [311] = −r6/10 + 117, [312]=[321]=r6/10 − 34, [322] = r6, [323] = [332] = −11r6/10 + 1378,[333] = 11r6/10 − 1107,where r6 ∈ {1010, 1020, . . . , 1170}.P r o o f. A simplification of formulas (3.1) taking into account the equalitiesS113(u, v, w) = S131(u, v, w) = S311(u, v, w) = 0. By Lemma 4, we have 1010 ≤ [322] = r6 ≤ 1170.Lemma 5. Let d(u, v) = d(u,w) = d(v,w) = 3. Then the following equalities hold :(1) [122] = −r17 + 336, [123] = [132] = r17, [133] = −r17 + 84;(2) [213] = [231] = r17, [212] = [221] = −r17 + 336, [222] = 39r17/4 + 3444,[223] = [232] = −35r17/4 + 1680, [233] = 31r17/4− 336;(3) [313] = [331] = −r17 + 84, [312] = [321] = r17, [322] = −35r17/4 + 1680,[323] = [332] = 31r17/4− 336, [333] = −27r17/4 + 522,where r17 ∈ {44, 48, . . . , 76}.P r o o f. A simplification of formulas (3.1) taking into account the equalitiesS113(u, v, w) = S131(u, v, w) = S311(u, v, w) = 0. By Lemma 5, we have 1015 ≤ [322] = −35r17/4 + 1680 ≤ 1295.The number d of edges between Σ(w) and Σ− ({w} ∪ Λ(w)) satisfies the inequalities359905 = 84 · 1010 + 271 · 1015 ≤ d ≤ 84 · 1170 + 271 · 1295 = 449225,267.786 ≤ 1343 − λ ≤ 334.245,1008.755 ≤ λ ≤ 1075.214,where λ is the mean value of the parameter λ(Σ).Shilla graphs with b = 5 and b = 6 57Lemma 6. Let d(u, v) = d(u,w) = 3 and d(v,w) = 2. Then the following equalities hold :(1) [122] = (−64r15 + 4r16 + 7364)/27, [123] = [132] = (64r15 − 4r16 + 1708)/27,[133] = (−64r15 + 4r16 + 560)/27;(2) [211] = −r15+20, [212] = [221] = (71r15+4r16+6392)/27, [222] = (−17r15−13r16+38311)/9,[223] = [232] = (−20r15 + 35r16 + 26095)/27, [233] = (64r15 − 31r16 + 8053)/27;(3) [311] = r15, [312] = [321] = (−71r15 − 4r16 + 2248)/27, [313] = (44r15 + 4r16 + 20)/27,[322] = (115r15 +35r16 +26716)/27, [323] = [332] = (−44r15 − 31r16+7297)/27, [333] = r16,where −10r15 + 4r16 + 20 is a multiple of 27, r15 ∈ {0, 1, . . . , 20}, and r16 ∈ {0, 1, . . . , 235}.P r o o f. A simplification of formulas (3.1) taking into account the equalitiesS113(u, v, w) = S131(u, v, w) = S311(u, v, w) = 0. By Lemma 6, we have998 ≤ [322] = (115r15 + 35r16 + 26716)/27 ≤ 1294.Let us count the number h of pairs of vertices y and z at distance 3 in the graph Ω, wherey ∈{uv31}, z ∈{uv32}.On the one hand, by Lemma 4, we have [323] = −11r6/10+1378, where r6 ∈ {1010, 1020, ..., 1170},therefore7644 = 8491 ≤ h ≤ 84267 = 22428.On the other hand, by Lemma 6, we have [313] = (44r15 +4r16 +20)/27, where r15 ∈ {0, 1, ..., 20},r16 ∈ {0, 1, ..., 235}, therefore7644 ≤∑i(44ri15 + 4ri16) + 995.55 ≤ 22428,6648.44 ≤∑i(44ri15 + 4ri16) ≤ 21432.45,4.946 ≤∑i(11ri15 + ri16)/1344 ≤ 15.947.If r15 = 0, then r16 + 5 is a multiple of 27 and r16 = 22.49, ....If r15 = 1, then 2r16 + 5 is a multiple of 27 and r16 = 11.38, ....In any case,∑i(11ri15 + ri16)/1344 ≥ 22,a contradiction.Theorem 3 is proved. ConclusionThe following are the main steps in creating a theory of Shilla graphs:(1) finding a list of feasible intersection arrays of Shilla graphs with b = 6;(2) classification of Q-polynomial Shilla graphs with b2 = c2.58 Alexander A. Makhnev and Ivan N. BelousovREFERENCES1. Belousov I. N. Shilla distance-regular graphs with b2 = sc2. Trudy Inst. Mat. i Mekh. UrO RAN, 2018.Vol. 24, No. 3. P. 16–26. (in Russian) DOI: 10.21538/0134-4889-2018-24-3-16-262. Brouwer A. E., Cohen A.M., Neumaier A. Distance-Regular Graphs. Berlin, Heidelberg: Springer-Verlag,1989. 495 p. DOI: 10.1007/978-3-642-74341-23. Coolsaet K., Jurǐsić A. Using equality in the Krein conditions to prove nonexistence of cer-tain distance-regular graphs. J. Combin. Theory Ser. A, 2008. Vol. 115, No. 6. P. 1086–1095.DOI: 10.1016/j.jcta.2007.12.0014. Gavrilyuk A. L., Koolen J.H. A characterization of the graphs of bilinear d × d-forms over F2. Combi-natorica, 2019. Vol. 39, No. 2. P. 289–321. DOI: 10.1007/s00493-017-3573-45. Jurǐsić A., Vidali J. Extremal 1-codes in distance-regular graphs of diameter 3. Des. Codes Cryptogr.,2012. Vol. 65, No. 1. P. 29–47. DOI: 10.1007/s10623-012-9651-06. Koolen J.H., Park J. Shilla distance-regular graphs. European J. Combin., 2010. Vol. 31, No. 8. P. 2064–2073. DOI: 10.1016/j.ejc.2010.05.0127. Makhnev A.A., Belousov I. N. To the theory of Shilla graphs with b2 = c2. Sib. Electr. Math. Reports,2017. Vol. 14. P. 1135–1146. (in Russian) DOI: 10.17377/semi.2017.14.097

Shilla Graphs with B=5 and B=6

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Shilla Graphs with B=5 and B=6

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