Table Of Content

The distance-3 graph of the Biggs-Smith graph 1 1 0 Italo J. Dejter 2 University of Puerto Rico t Rio Piedras, PR 00936-8377 c O [email protected] 0 1 Abstract O] The distance-3 graph S3 of theBiggs-Smith graph S is shown tobe: (a) a connected edge-disjoint union of 102 tetrahedra (copies of K4) and as C such the K4-ultrahomogeneous Menger graph of a self-dual (1024)-confi- h. guration; (b) a union of 102 cuboctahedra, (copies of L(Q3)), with no t twosuchcuboctahedrahavingacommonchordless4-cycle;(c)notaline a graph. Moreover,S3 isshowntohaveaC-ultrahomogeneouspropertyfor m C = {K4}∪{L(Q3)} restricted to preserving a specific edge partition of [ L(Q3)into2-paths,witheachtriangle(resp. edge)sharedbytwocopiesof 2 L(Q3)plusoneofK4(resp. fourcopiesofL(Q4)). Boththedistance-2and distance-4 graphs, S2 and S4, of S appearin thecontextassociated with 2 v the above mentioned edge partition. This takes us to ask whether there 8 anynon-line-graphicalconnectedK4-ultrahomogeneousMengergraphsof 8 self-dual (n4)-configurations that are edge-disjoint unions of at least two 5 copies of K4 for positive integers n∈/ {42,102}. 0 . 2 1 Introduction 0 0 1 Given a (di)graph Γ and a positive integer k diameter(Γ), the distance-k ≤ : graph Γk of Γ has V(Γk) = V(Γ) and an arc in Γk from a vertex u to a vertex v i v =uwheneverthereisashortestk-arcoflengthk inΓfromutov. Ak-arcin X 6 a graph is a sequence of vertices (v ,...,v ) such that consecutive vertices are 0 k r adjacent and v = v when 0 < i < k [9]. A k-arc can also be interpreted a i−1 6 i+1 as a directed walk of length k in which consecutive edges are distinct [11]. Ultrahomogeneous(or UH) graphs were introducedand treated in [3, 8, 10, 14, 15], but we deal here with the following modified concept of ultrahomogeneity. Given a collection of (di)graphs closed under isomorphisms, a (di)graph G is C said to be -UH if every isomorphism between two induced members of in G C C extends to an automorphism of G. If is the isomorphism class of a graph H, C thenG is saidto be H -UHorH-UH.If is the isomorphismclass ofagraph { } C H with an edge partition Ω into 2-paths), then G is said to be Ω-preserving H -UHorH-UHifonlyeveryΩ-preservingisomorphismbetweentwoinduced { } copiesofH extends toanautomorphismofG. In[12], -UHgraphsaredefined C 1 and studied when is the collection formed either by the complete graphs, or C the disjoint unions of complete graphs, or the complements of those disjoint unions. Let M be a subgraph of a graph H and let G be both an M-UH and an H- UH graph. (A particular case has H = L(Q ) with an edge partition Ω into 3 2-paths, as for example the L(Q ) represented in Figure 4, Section 4 below, 3 with M equal to a 2-path in Ω). We say that G is an (Ω-preserving) H - M { } UH graph if, for each induced copy H of H in G containing an induced copy 0 M of M, there exists exactly one induced copy H = H of H in G with 0 1 0 6 V(H ) V(H ) = V(M ) and E(H ) E(H ) = E(M ). These vertex and 0 1 0 0 1 0 ∩ ∩ edge conditions can be condensed as H H = M . We say that such a G is 0 1 0 ∩ tightly fastened. This is generalizedby saying that an H -UH graphG is an M { } ℓ-fastened H -UH graph if given an induced copy H of H in G containing M 0 { } an induced copy M of M, then there exist exactly ℓ induced copies H = H 0 i 0 6 ofH inGsuchthatH H M ,foreachi=1,2,...,ℓ,withH H =M . i 0 0 1 0 0 ∩ ⊇ ∩ Observe that in these definitions we started by assuming that G is M-UH and H-UH. Atransformationofdistance-transitivegraphsinto -UHgraphsthatgoesfrom C theCoxetergraphon28verticesintotheKleingraphon56vertices[7]isapplied below to the Biggs-Smith graph [2, 4] seen as a C -UH graph. This S { 9}P4 allows to show that the graphobtained by the said transformation, namely the distance-3 graph 3 of , is a connected edge-disjoint union of 102 tetrahedra S S (copiesofK )aswellasaunionof102cuboctahedra,(copiesofL(Q ))withno 4 3 twosuchcuboctahedrahavingacommon4-hole(chordless4-cycle). Inaddition, 3 is shown to possess a -UH property for = K L(Q ) restricted to 4 3 S C C { }∪{ } preserving a specific edge partition of L(Q ) into 2-paths, with each triangle 3 (resp. edge) of 3 shared by two copies of L(Q ) plus one of K (resp. four 3 4 S copies of L(Q )) exactly. Moreover, 3 is seen to be the Menger graph [5] of a 3 S self-dual (102 )-configuration,but not a line graph. Both 2 and 4 are shown 4 S S to be related to the above mentioned edge partition. Thus, the connected edge-disjoint union of 42 copies of K in [6], forming a 4 X K -UH graph which is the Menger graph of a (42 )-configuration but is not a 4 4 linegraph,isnottheonlycaseofMengergraphofaself-dual(n )-configuration 4 which is connected edge-disjoint union of copies of K and K -UH but not a 4 4 line graph: 3 provides another such graph. This allows to raise the following S question. Question 1 Are there any non-line-graphical connected K -UH Menger graphs 4 of self-dual (n )-configurations that are edge-disjoint unions of at least two 4 copies of K , for positive integers n / 42,102 ? 4 ∈{ } We recall that the line graph of the d-cube, is K -UH, where 3 d Z [6]. d ≤ ∈ Other considerations related to Question 1 can be found in Section 5. 2 2 Preliminaries We recall that the Biggs-Smith graph has order n = 102, diameter d = 7, S girthg =9andautomorphismgroup =PSL(2,17)whichisalsoshowntobe A that of the distance-3 graph 3 of in Theorem7. Let k be the largestinteger S S s such that is s-arc transitive. Then k = 4. Finally, we also recall that the S number η of 9-cycles of is η =136. S Given a finite graph H and a subgraph M of H with V(H) >3, a graph Γ is | | strongly zipped (or SZ) H -UH if for each sequence of connected subgraphs M { } M =M ,M ...,M K such that M is obtained from M by the deletion 1 2 t 2 i+1 i ≡ ofavertex,fori=1,...,t 1,itholds thatG is a(2i 1)-fastened H -UH − − { }Mi graph, for i=1,...,t. Let P be a (k 1)-path. Let C be a cycle of length g. Theorem 1 below as- k g − sertsthat isSZ C -UH,namelySZ C -UH.Also, 3 isSZ H -UH S { g}Pk { 9}P4 S { }M (Theorems 6-7), where H = L(Q ) and M is a triangle, but some additional 3 concepts are needed: A graph G is rK -frequent if every edge e of G is inter- s section in G of exactly r induced copies of K , and these copies have only e s and its ends in common. (For example, K is 2K -frequent, and L(Q ) is 1K - 4 3 3 3 frequent). A graph G is H ,H -UH, where H is iK -frequent, (i = 1,2), { 2 1}K3 i 3 if: (a)GisanH -UHgraphandanedge-disjointunionofinducedcopiesofH ; 2 2 (b) there exists an edge partition Ω of H into 2-paths, and G is Ω-preserving 1 SZ H -UH;(c)eachinducedcopyofH inGhaseachofitsinducedcopies { 1}K3 2 of K in common with exactly two induced copies of H in G. 3 1 Properties of needed in the rest of the paper are covered in Section 3. In S Section 4, consideration of the distance-3 graphs of the 9-cycles of yields a S K ,L(Q ) -UH graph via zipping (meaning identification into an edge) { 4 3 }K3 Y of corresponding oppositely oriented arcs of the resulting 3-cycles. Theorem 7 uses Theorem2, that establishes anexplicit 1-1correspondencebetween the 18 disjoint unions of 4 17-cycles each in and the points of the projective line of S thefieldGF(17). Theorem8,establishing = 3,alsoyieldstwoothergraphs, Y S appearing in the way the 102 copies of L(Q ) intersect, namely 2 and 4. 3 S S 3 Properties of the Biggs-Smith graph S Let A = (A , A , ..., A ), D = (D , D , ..., D ), C = (C , C , ..., C ), 0 1 g 0 2 f 0 4 d F = (F , F , ..., F ) be 4 disjoint 17-cycles. Each y = A,D,C,F has vertices 0 8 9 y with i expressed as an heptadecimal index up to g = 16. We assume that i i is advancing in 1,2,4,8 units mod 17, stepwise from left to right, respectively for y =A,D,C,F. Thenthe Biggs-Smithgraph is obtainedby adding to the S disjointunion A D C F, for eachi Z , a 6-vertextree T formedby the 17 i ∪ ∪ ∪ ∈ edge-disjoint union of paths A B C , D E F , and B E , where the vertices A , i i i i i i i i i D , C , F arealreadypresentinthe cyclesA, D, C, F,respectively,andwhere i i i 3 the verticesB and E are new and introducedwith the purpose ofdefining the i i tree T , for 0 i g = 16. A corresponding Frucht diagram [13] of via i ≤ ≤ S Z is depictedonthe left ofFigure 2below,withnonzeroFruchtnotationover 17 the loops, and null Frucht notation over the remaining edges. Now, has the S collection of 9-cycles formed by: 9 C S0=(A0A1B1C1C5C9CdC0B0), W0=(A0A1B1E1F1F9F0E0B0), T0=(C0C4B4A4A3A2A1A0B0), X0=(C0C4B4E4D4D2D0E0B0), U0=(E0F0F9F1FaF2E2D2D0,) Y0=(E0B0A0A1A2B2E2D2D0), V0=(E0D0D2D4D6D8E8F8F0), Z0=(F0F8E8B8C8C4C0B0E0), and those 9-cycles obtained from these, as Sx,...,Zx, by uniformly adding x Z mod 17 to all subindexes i of vertices y , so that =136. 17 i 9 ∈ |C | An auxiliary table presenting the form in which the 9-cycles above share the vertexsetsof3-arcs,eitheroppositelyorientedornot,isshownbelow. Thetable details,foreachoneofthe9-cyclesξ =S0,...,Z0,(expressedasξ =(ξ ,...,ξ ) 0 8 in the shownvertexnotation), each9-cycle η in Si,...,Zi;i=0,...,16 ξ { }\{ } intersecting ξ in the succeeding 3-paths ξ ξ ξ ξ , for i = 0,...,8, with i i+1 i+2 i+3 additionsinvolvingitakenmod9. Eachsuchη inthe auxiliarytable haseither a preceding minus sign, if the corresponding 3-arcs in ξ and η are oppositely oriented, or not, otherwise. Each shown η (resp. η ) in the table has the j j − subindex j indicating the equality of initial vertices η =ξ (resp. η =ξ ) of j i+3 j i those 3-arcs, for i=0,...,8: S0:(-Te,T1,-Z1,Sd,S4,-Z9,Td,-T0,U0), W0:(-U0,W8,W9,-U1,X2,-Xb,Y0,-X0,X9), 1 7 4 4 3 3 0 6 8 4 2 1 3 7 4 6 8 5 T0:(S64,-S03,-Y22,T41,T3g,-Y10,-S70,S1g,V80), X0:(-V40,X2f,X12,-V34,-W56,W88,-Z80,W4f,-W70), U0:(Yg,-U1,Z1,-Wg,-W0,Z9,-Ug,Y0,S0), Y0:(U0,-T0,-Tf,U1,-Y2,Vf,W0,V0,-Yf), 3 6 7 3 0 0 1 0 8 7 5 2 0 8 2 6 5 4 V0:(-Zd,-V4,Y2,-Xd,-X0,Y0,-Vd,-Z0,T0), Z0:(U8,-Z8,-V4,-S8,-Sg,-V0,-Z9,Ug,-X0), 2 6 5 3 0 7 1 5 8 5 6 0 5 2 7 1 2 6 This table is extended to all of by adding x Z uniformly mod 17 to all 9 17 C ∈ superindexes. Theorem 2 is an SZ C -UH graph, for i = 0,1,...,k 2 = 2. In S { 9}Pi+2 − particular, is a C -UH graph and has exactly 2k 23n9 1 =136 9-cycles. S { 9}P4 − − Proof. We have to see that is a (2i+1 1)-fastened C -UH graph, for S − { 9}P4−i i=0,1,2. Infact,each(3 i)-pathP of issharedexactlyby 2i+1 9-cycles 4 i − − S of , for i = 0,1,2. For k = 4, any edge (resp. 2-path, resp. 3-path) of G is S shared by 8 (resp. 4, resp. 2) 9-cycles of G. This means that a 9-cycle C of G 9 sharesa P (resp. P , resp. P ) with exactly other 7 (resp. 3, resp. 1)9-cycles. 2 3 4 The statement follows from this and the details previous to it together with a simple counting argument for the number of 9-cycles. The reconstruction of from the 17-cycles A,D,C,F will be denoted S (A,D,C,F)= (P1,P2,P4,P8), ℵ ℵ where P1 = A, P2 = D, P4 = C, P8 = F, in order to see that it yields 8 au- tomorphisms of obtained by multiplying modulo 17 the successive exponents S 4 of P, (=P1 =A), by 2: (P1,P2,P4,P8) (P2,P4,P8,P−1) ... (P−8,P1,P2,P4) (P1,P2,P4,P8). ℵ →ℵ → →ℵ →ℵ Observethata9-cyclexof sharingexactlya1-pathx withAsharesexactly A S a2-pathx withF,withdistance3betweenx andx realizedbytwovertex- F A B disjoint 3-paths in x. Note that x ,x and the two 3-paths separating them A F are edge-disjoint. By using repeatedly this fact, it can be seen , say via a representationof showingtheverticesofA,D,C,F symmetricallydistributed S onfourconcentriccircles,thatthereisanotherreconstruction (A0,D0,C0,F0) ℵ of with four disjoint 17-cycles: S A0=(A0A1B1C1C5B5E5F5FdF4FcEcBcCcCgBgAg)= (A00,A01,...,A0g), D0=(F0F8E8D8D6E6B6C6C2CfCbBbEbDbD9E9F9)=(D00,D10,...,Dg0), C0=(C0C4B4A4A3B3E3D3D1DgDeEeBeAeAdBdCd)=(C00,C10,...,Cg0), F0=(D0D2E2F2FaEaBaAaA9A8A7B7E7F7FfEfDf)=(F00,F10,...,Fg0), where A = A0, F =D0, C =C0, D = F0, etc., and where the vertices A0, 0 0 0 0 0 0 0 0 i D0, C0, F0, for each fixed i Z , are the degree-1 vertices of a 6-vertex tree i i i ∈ 17 T0, like the tree T above, formed by the disjoint union of the paths A0B0C0, i i i i i D0E0F0 and B0E0. An involution for i Z in (A0,D0,C0,F0) is also i i i i i ∈ 17 ℵ present whose orbits are y0 , y1,yg , y2,yf , y3,ye , y4,yd , y5,yc , { } { } { } { } { } { } y6,yb , y7,ya and y8,y9 , where y A,B,C,D,E,F . For example, { } { } { } ∈ { } (A,D,C,F) and (A0,D0,C0,F0) are represented in Figure 1 below, where ℵ ℵ A,A0 (resp. D,D0;C,C0;D,D0havethickblack(resp. green;blue;red)edges. Figure 1: Assemblies (A,D,C,F) and (A0,D0,C0,F0) of ℵ ℵ S There is a copy of the dihedral group D as a subgroup of the automorphism 8 group of that has: (a) a cyclic subgroup of order 8 generated by an au- A S tomorphism that sends A onto D (given below by a matrix ρ) and (b) an i 2i automorphism that sends A onto A0 (given below by a matrix α2), where i i i Z . Moreover, contains a subgroup which is a semidirect product of 17 0 ∈ A A this copy of D and a copy of Z generated by a matrix θ, also given below. 8 17 5 There are 18 reconstructions of , of which we just treated (A,D,C, F) and S ℵ (A0,D0,C0,F0). The remaining sixteen reconstructions, denoted (Aj,Dj, ℵ ℵ Cj, Fj), have 17-cycles Aj,Dj,Cj,Fj, with 0 = j Z , obtained from 17 6 ∈ A0,D0,C0, F0 via uniform addition of indexes mod 17. If we denote by ζ A| the order-9 subgroup of generated by an adequate rotation ζ of a 9-cycle of A , then the 18 reconstructions above are related by means of the subgroups S ζ. Eachsuchsubgroup ζ of : (a)hasasemidirectproductwith equal 0 A| A| A A to , showing that = 17 32 24 and how is structured by means of A |A| × × A semidirect products from D , Z and ζ; (b) partitions V( ) into eleven 9- 8 17 A| S orbits and one 3-orbit . The stabilizer in ζ of any of the three vertices of ζ O A| V( ) (on which ζ acts) contains a copy of the symmetric group ζ O ⊂ S A| ⊂ A S . Each one of these three vertices have distance 3 from ζ realized by paths 3 whose ends in ζ subdivide it into paths of length 3. (This disposition of vertices will yield in Section 4 three disjoint copies of K in ). For example, if 4 Y ζ =(F F F F F E D D E ), then =(A D D ). g 8 0 9 1 1 1 g g ζ 0 7 a O 1 A D 2 7 ✎☞ ✎☞ ✬ 6 ✩ ss s ✍B ✌ ✍E ✌ ✤ 1 ✜ 7 ✗4 ✔ s s ☛ ✟ ☛ ✟ 4 C F 8 1 ✎☞ ✎☞ ✎0☞(9)1(9) 2(9) 3(9) 4(9) 5(9) 6(9) 7(9) 8(9) 9(9) a(9) b(3) s s s s s s s s s s s s s s ✍✌ ✍✌ ✍✌ Figure 2: Frucht diagrams of via Z and Z 17 9 S There is an11-arcdepartingfrom eachvertex ofζ, yielding a totalof 9 disjoint 11-arcs, three arriving at each vertex of , with component vertices at each ζ O fixed distance d < 11 from ζ forming a corresponding 9-orbit d, so we can O write 0 = ζ and 12 = . A corresponding Frucht diagram [13] of via ζ O O O S Z is depicted on the right of Figure 2, where each number denoting a vertex 9 is followed with the parenthesized size of the Z -orbit it represents, and with 9 nonzero Frucht notation[13] overedges taken as arcs from left to right, so that the corresponding oppositely oriented arcs have the respective negative Frucht notation;horizontaledges,nothavinganexplicitFruchtnotationdenoted,must be taken with null Frucht notation. There is a correspondence between the originalnotationof andanew onebasedonthisFruchtdiagram,obtainedby S re-expressing the 9 11-arcs as follows: FFFFFEDDg809111gEEEFFBDDa780913eFFBDBCEEf289013eFEACDCBFbf98e2e3FDCABEFC35f96e23FBAAEDBCc5a6e2d7FABAACCE456bd2bdFAABCACBd46b6cfdFAEACBBC53ba7cEfdADBEBAC52b7ca0fDADDEEBA5179ca0gDADDDDAA7077a0a0 →→→→→→→→ 0000000010325476111111111032547622222222103254763333333310325476444444441032547655555555103254766666666610325476777777771032547688888888103254769999999910325476aaaaaaaa10325476bbbbbbbb10022110 EgBgCgCcC8C4B4E4D4D5D8Da → 08182838485868788898a8b2 The 18 reconstructions of cited above correspond in a 1-1 fashion to the 18 S points of the projective line over the field GF(17); this 1-1 correspondence P 6 is to be set by the end of this section. Elements of PSL(2,17), represented by 2 2-matricesoverGF(17),actoverthesepointsbyleftmultiplication. Inorder × to fitwith the 2 2-matricesprovidedbelow, (A,D,C,F) mustcorrespondto the point of g×iven by the column vector (cid:0)1(cid:1)ℵ. Then the point at , given by P 9 ∞ (cid:0)0(cid:1),andtheremainingpoints,givenbyvectorcolumns(cid:0)1(cid:1),for9=j GF(17), 1 j 6 ∈ correspond to the remaining reconstructions (Ai,Di,Ci,Fi), for i GF(17). ℵ ∈ The stabilizerof the groupPSL(2,17)atanyfixed elementof coincides with P theautomorphismgroupofthe3-cubeQ . Toseethatthisisalsothe stabilizer 3 of at the vertex A of , recall that the number of 4-arcs departing AA0 A 0 S from a fixed vertex of is 24. Each such 4-arc determines a unique oriented S 17-cycleandauniqueoriented9-cycleof . Thus,therearetwentyfouroriented S 17-cycles passing through each vertex of , in particular through A . Two of 0 S these oriented 17-cycles are A = (A A A ...A ) and its oppositely oriented 0 1 2 g 17-cycle, A. The other oriented 17-cycles through A are obtained from the 0 − reconstructions of above. In fact, has generators α,β,γ of orders 4, 2 S AA0 and 3 respectively, that map the successive vertices of A, as presented above, onto the respective successive vertices of D9,D10, C3. Moreover, α,β,γ − − form a generating set of the automorphism group of Q . For example, the 3 automorphismthatsendsAonto Aequalsα2βγ2. Likewise,Aissentontothe − following17-cycles,bymeansoftheautomorphismsexpressedinthesubsequent contiguous line: A0 A1 C3 C4 D7 D8 D9 Da Cd Ce Ag α2 α3γ α3γ2 αγ αγ2 α3 α3βγ2 β α2βγ α2β βγ A0 A1 C3 C4 D7 D8 D9 Da Cd Ce Ag β−γ2 −α2γ2 −γ −γ2 α−2γ α−βγ2 −α α−βγ α−β α−3βγ α−3β completing the specifications for the 24 elements of , that yield informa- AA0 tion about 12 of the 18 reconstructions of , with each two oppositely oriented S 17-cycles displayed belonging to a specific reconstruction. These twelve reconstructions are respectively (Ai,Di,Ci,Fi), for i=0,1,3,4,7,8,9,a,d,e,g. ℵ The generators α,β,γ of can be associated with the following matrices: AA0 α → (−01 61), β → (−01 01), γ → (−01−11). These matrices allow to establish a group embedding of onto PSL(2,17). AA0 ByLagrange’stheorem, partitions into102classes,correspondingtothe AA0 A 102 vertices of . Also, γ has as cubic roots the following three matrices: S (√3γ)1 → (92 181), (√3γ)2 → (62 185), (√3γ)3 → (96 1115). Any of these cubic roots can be associated to a rotation generator of ζ. Let A| (√3γ)1 be such a generator. We can take the copies of the subgroups Z17 and D in above to be generated in PSL(2,17) by its elements represented by 8 A θ → ( 08 22), ρ → (−01 81), α2 → (−−16 61), 7 whereθ hasorder17,ρ hasorder8 andρ combineswithα2 (orwithβ) toform a copyofD in . These copiesofthe two subgroupsZ andD in , related 8 17 8 A A adequatelyviasemidirectproductstothecopyofZ9 generatedby(√γ)1,yields an isomorphism PSL(2,17) to be used specifically in Theorem 7. A→ BecauseAand Aarerepresentedbytheidentityandbyα2βγ2,respectively,it − can be seen that the reconstruction (A,D,C,F) of has associatedthe point (cid:0)19(cid:1) of P with the vertex A0. Similaℵrly, because A0Sand −A0 are represented by α2 and by βγ2, respectively, the reconstruction (A0,D0,C0,F0) of has associatedthepoint(cid:0)110(cid:1)ofP withthevertexA0. ThℵeselectionofθismotiSvated forittakesthispoint,(cid:0)1(cid:1) ,viasuccessivepowersθ,θ2,...ontothefollowing 10 ∈P points: (1),(1),(1),(1),(1),(1),(1),(1),(0),(1),(1),(1),(1),(1),(1),(1),(1), 10 15 16 14 11 6 13 0 1 1 5 12 7 4 2 3 8 which correspond to the successive reconstructions (Ai, Di, Ci, Fi) of , for ℵ S i = 0,...,8,10,...,16, thus establishing the desired 1-1 correspondence from the family of 18 reconstructions of onto . S P Theorem 3 In , there exist 18 disjoint unions of four 17-cycles each, deter- S mining 18 corresponding reconstructions of 3. Those unions in are in 1-1 S S correspondence with the 18 points of the projective line of GF(17) in such a P way that the action of on the family of these reconstructions is equivalent to A the natural action of PSL(2,17) on . P 4 The K ,L(Q ) -UH graph = 3 4 3 K { } 3 Y S We keep using the notation of and its collection of 136 9-cycles in Section 9 S C 3. Let ( )3 be the collection of distance-3 graphs of 9-cycles of . In each 9 9 C C copy w w of P in a member C3 = (C )3 of ( )3, where C , let us 0 1 2 9 9 C9 9 ∈ C9 denotebyetheedgebetweenw andw . Then,theinitialvertexw ,theinitial 0 1 0 flag w ,e , the terminal flag e,w and the terminal vertex w of w w are 0 1 1 0 1 { } { } indicated,respectively,bymeansofthenamesoftheverticesv ,v ,v ,v ofthe 0 1 2 3 copy v v v v of P in C for which w w stands in C3. 0 1 2 3 4 9 0 1 9 For example, if C = S0 = (A A B C C C C C B ), so that C3 = (S0)3 = 9 0 1 1 1 5 9 d 0 0 9 (A C C )(A C C )(B C B ), then the initial flag of the copy A C of P in 0 1 d 1 5 0 1 9 0 0 1 2 C3 = (S0)3 is indicated by A , the terminal flag by B , while A and C 9 1 1 0 1 are indicated by themselves, namely A and C . We get the indications over 0 1 C3 =(S0)3 shown in Figure 3. 9 We zip now corresponding arc pairs of the distance-3 graphs C3 obtained from 9 , in order to get a graph with the desired -UH properties, meaning that S Y C 8 B0 C0 A0 B0 A1 A0 ✎ ☎ ✎ ☎ ✎ ☎ A0 A1 B1 C1 C5 C9 Cd A1 B1 C1 C5 C9 Cd C0 B1 C1 C5 C9 Cd C0 B0 ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ Figure3: ExampleofindicationsoverC3 =(S0)3 =(A A B C C C C C B )3 9 0 1 1 1 5 9 d 0 0 we identify into an edge corresponding oppositely oriented arcs of the resulting 3-cycles. Thefollowingsequenceofoperationsisperformed;(comparewith[7]): ( )3 . (1) 9 9 S → C → C → Y Next, we explain how this operation is composed. The distance-3 S → Y graphsC3 ofthe1369-cyclesC of areformedbythreedisjoint3-cycleseach, 9 9 S yielding a total of 3 136 = 408 3-cycles. Thus, determines a collection of 9 × C 408 triangles in , with each edge of shared by exactly two such triangles. Y Y In fact, we identify or zip in these triangles the pairs of copies of P obtained 2 as distance-3 graphs of copies of P in . This yields 102 copies of K that 4 4 S can be subdivided into six subcollections Yi of 17 copies of K each, where 4 { } Y A,B,C,D,E,F andi 0,1,...,16=g =Z . ThevertexsetsV(Yi), 17 ∈{ } ∈{ } each followed by the set Λ(Y ) of copies of K containing the corresponding i 4 vertex Y (as in the notation of Section 3), can be taken as follows, showing i Z -symmetries produced by the change-of-signinvolution in Z : 2 17 VVVVVV((((((FEDCABiiiiii))))))======{{{{{{CCCADAiiiiii+,,,+,63,, CFDFDDiiiiii,,−−,,63,, ABEBEFiiiiii++++++514782,,,,,, BAEBEFiiiiii−−−−−−514872}}}}}},,,,,, ΛΛΛΛΛΛ((((((EFDCABiiiiii))))))======{{{{{{CACCDAiiiiii,+,,+,12,, CFDFDDiiiiii−,,−,,12,, BABEFEiiiiii++++++486753,,,,,, BABEFEiiiiii−−−−−−486753}}}}}},,,,,, wherei Z . Thisrevealsaduality mapφ fromthe 102verticesof ontothe 17 ∈ S 102 copies of K . In fact, the copies of K are the vertices 4 4 φ(A )=A3i =A , φ(B )=B 7i =B , φ(C )=C3i =C , i ∗i i − i∗ i i∗ φ(D )=D5i =D , φ(E )=E6i =E , φ(F )=F5i =F , i i∗ i i∗ i i∗ (i Z ), of a graph φ( ) = with a structure similar to that of 17 ∗ ∈ S S ≡ S the vertices A ,B ,C ,D ,E ,F of and whose copies of K can be precisely i i i i i i 4 S denotedy =A ,B ,C ,D ,E ,F ,withcorrespondingvertexsetsΛ(y )asspec- i i i i i i i i ified above. Moreover, φ: is a graph isomorphism, with the adjacency ∗ S →S of similar to that of . ∗ S S The distance-3graphofeach9-cycleof iscomposedby three disjoint3-cycles S in . For example, the distance-3 graphs of S0,...,Z0 are: Y WTSUV00000→→→→→{{{{{EECAC7b140\\\\\ACACEe541g=====(((((CAEEA00000,,,,,ACDFE41141,,,,,ACEEFd1208))))),,,,,CEEBB1ab53\\\\\EADCF24gf8=====(((((AACFD11040,,,,F,CFAD1a53,6,,,ECD,AF00208)))),),,,,FFBDB99e25\\\\\CFDDF99ab2=====(((((BBDBF11942,,,,,FFCAD92928,,,,B,DBBF00000)))))}}}}};;;;; YX00→→{{CA10\\FE1d==((EC00,,AE14,,ED20)),,DA42\\EB84==((CB40,,DA42,,ED02)),,DD02\\BAf2==((AB04,,BD22,,DB00))}};; Z0→{F0\B9=(F0,B8,C0), F8\Bg=(F8,C8,B0), A4\D4=(E8,C4,E0)}. 9 This way, it can be seen that is a K -UH graph. Moreover, the vertices and 4 Y copies of K in are the points and lines of a self-dual (102 )-configuration, 4 4 Y which in turn has as its Menger graph. However, in view of Beineke’s char- Y acterization of line graphs [1], and observing that contains induced copies of Y K , which are forbidden for line graphs of simple graphs, we conclude that 1,3 Y is non-line-graphical. We obtain the following statement. Theorem 4 is an edge-disjoint union of 102 copies of K , with four such 4 Y copies incident to each vertex. Moreover, is a non-line-graphical K -UH 4 Y graph. Its vertices and copies of K are the points and lines, respectively, of a 4 self-dual(102 )-configurationwhichhas precisely asitsMenger graph. Inpar- 4 Y ticular, isarc-transitivewithregulardegree12,diameter 3anddistancedistri- Y bution(1,12,78,11). ItsassociatedLevigraph[5]isa2-arc-transitivegraphwith regular degree 4, diameter 6 and distance distribution (1,4,12,36,78,62,11). Proof. The statement is immediate from the data given by the construction of via operation (1) and because of the fact that is a distance-transitive Y S graph. Remark 5 A proof yielding the group structure of the automorphism group B of is part of the proof of Theorem 6. However, the Magma software allowed Y to verify that the automorphism group of is = = PLS(2,17), whose Y B A order is 2448, and that the Levi graph associated to has automorphism group Y SL(2,17), whose order is 4896. Theorem 6 is a K -UH graph. 4 Y { } Proof. Consider an isomorphism Ψ : Θ Θ between two copies Θ ,Θ of 1 2 1 2 K in . By construction, each Θ (i =→1,2) arises from 4 9-cycles θj in 4 Y i i S (j =1,2,3,4) related so that each one of them have their union as a subgraph Θ of with 4 vertices vj of degree 3 and 12 vertices of degree 2 which are i S i the internal vertices of 6 3-paths whose endvertices are the vertices vj. For i example, the vertices v1 = B ,v2 = B ,v3 = F ,v4 = C ,v1 = B ,v2 = 1 0 1 1 1 9 1 9 2 1 2 B ,v 13 =F ,v4 =C in determine such subgraphs Θ ,Θ in and Θ ,Θ in2 .2Clearlya, Ψ2induaces aSn isomorphism Ψ : Θ Θ t1hat2sendYs say eac1h v2j S 1 → 2 1 onto a corresponding vj (j = 1,2,3,4). An automorphism Ψ of exists that 2 S extends Ψ. Moreover,Ψ determines an automorphism of that restricts to Ψ, Y which completes our verification that is a K -UH graph. 4 Y { } Eachofthe102copiesofK in arisesfromthedistance-3graphsoffourofthe 4 Y 1369-cyclesof . Thesubgraphof spannedbythesefour9-cyclescontainsfour S S degree-3vertices,(whichareinitialandterminalverticesofcorrespondingcopies of P , as in the first paragraph of the section), and twelve degree-2 vertices, 4 (internal vertices in those copies of P ). These twelve vertices form a copy of 4 L L(Q ) in . For the copy A0 of K in , the copy = a0 of L(Q ) in can 3 4 3 Y Y L Y 10