It is known that the five non-isomorphic quasi-symmetric 2-(31, 7, 7) designs lead to non-isomorphic strongly regular graphs with parameters (155, 42, 17, 9). We will show that there exist no isomorphisms among these graphs and the block graphs of the Steiner triple systems STS(31) except the isomorphism between the block graphs of the point-plane design and the point-line design of PG(4, 2)

Horiguchi  Naoyuki

Nakasora  Hiroyuki

Wakabayashi  Takehisa

Yamagata University Academic Repository

Bull. of Yamagata Univ., Nat. Sci., Vol.16, No.1, Feb. 2005On the strongly regular graphs obtained fromquasi-symmetric 2-(31, 7, 7) designsNaoyuki HORIGUCHI∗, Hiroyuki NAKASORA†and Takehisa WAKABAYASHI‡(Received May 26, 2004)AbstractIt is known that the ﬁve non-isomorphic quasi-symmetric 2-(31, 7, 7)designs lead to non-isomorphic strongly regular graphs with param-eters (155, 42, 17, 9). We will show that there exist no isomorphismsamong these graphs and the block graphs of the Steiner triple sys-tems STS(31) except the isomorphism between the block graphs ofthe point-plane design and the point-line design of PG(4, 2).1 IntroductionA t-(v, k, λ) design is a pair (X,B), where X is a set of “points” of cardi-nality v, and B is a collection of k-element subsets of X called “blocks”, withthe property that any t points are contained in precisely λ blocks. An inter-section number is the number of points contained in two blocks. A 2-designis called quasi-symmetric if the intersection number of the design takes justtwo values. A graph consists of a ﬁnite set V of “vertices” together with a setE of “edges”, where an edge is a subset of a vertex set of cardinality 2. Twographs (V,E), (V �, E�) are isomorphic if there is a bijection φ : V → V � suchthat (v, w) ∈ E if and only if (φ(v), φ(w)) ∈ E �. An automorphism of a graphis an isomorphism from the graph to itself. The set of all automorphisms ofa graph forms a group and it is called the automorphism group. A stronglyregular graph with parameters (n, k, λ, µ) is a graph with n vertices in which∗Graduate School of Science and Technology, Chiba University, Chiba 263–8522, Japan†Graduate School of Natural Science and Technology, Okayama University, Okayama700–8530, Japan‡Graduate School of Science and Engineering, Yamagata University, Yamagata 990–8560, Japan1Naoyuki HORIGUCHI, Hiroyuki NAKASORA, and Takehisa WAKABAYASHIthe number of common neighbors of x and y is k, λ or µ according as x andy are equal, adjacent or non-adjacent respectively.Let D = (X,B) be a quasi-symmetric design with the intersection num-bers λ1, λ2 (λ1 < λ2). The block graph of D is a graph whose vertices are theblocks of D and whose edges are the pairs (Y, Z) of blocks with |Y ∩Z| = λ2.Then it is known that the block graph is strongly regular (cf. [2, Theorem5.3], [5, Theorem 37.7]).Tonchev [8] showed that there are exactly ﬁve non-isomorphic quasi-symmetric 2-(31, 7, 7) designs. Five strongly regular graphs with parame-ters (155, 42, 17, 9) are obtained from the ﬁve designs. Then Stoichev [7]proved that the ﬁve graphs are non-isomorphic by comparing the automor-phism groups. On the other hand, strongly regular graphs with the sameparameters are also obtained from the Steiner triple systems STS(31). It isknown that if v ≥ 15, non-isomorphic Steiner triple systems STS(v) lead tonon-isomorphic strongly regular graphs [1]. In this paper, we consider thecliques of maximum size in the ﬁve graphs obtained from the quasi-symmetricdesigns by using Magma. We will give an alternative proof of Theorem inStoichev [7], and further we will prove the following theorem:Theorem 1. There exist no isomorphisms among the block graphs from thequasi-symmetric 2-(31, 7, 7) designs and the Steiner triple systems STS(31)except the isomorphism between the block graphs of the point-plane designand the point-line design of PG(4, 2).2 Quasi-symmetric designs and strongly reg-ular graphsWe begin this section with Tonchev’s result [8]. Conway and Pless [3]showed that there exist exactly ﬁve inequivalent binary extremal doubly-even self-dual [32, 16, 8] codes. We denote these codes by C1, C2, C3, C4 andC5, whose components are q32, r32, 2g16, 8f4 and 16f2, respectively [3, TableIII]. The codewords of weight 8 in Ci form a 3-(32, 8, 7) design Di by theAssmus–Mattson theorem [2, Theorem 14.11]. As a derived design of Di, aquasi-symmetric 2-(31, 7, 7) design D�i is obtained. Its intersection numbersare 1 and 3. In addition, all derived designs of Di are isomorphic since theautomorphism group of Di acts transitively on the points. It is shown in [8]that the ﬁve quasi-symmetric 2-(31, 7, 7) design D�i (i = 1, 2, . . . , 5) obtainedfrom Di are non-isomorphic.Let Gi be the block graph of D�i (i = 1, 2, . . . , 5). The following theoremis proved by Stoichev [7].2On the strongly regular graphs obtained from quasi-symmetric 2-(31, 7, 7) designsTheorem 2. The ﬁve graphs G1, G2, G3, G4, G5 are non-isomorphic.We will give an alternative proof of the theorem. The generator matrixof the code Ci is obtained from Sloane [6]. We set the generator matrix inMagma as follows:m := KMatrixSpace(GF(2),16,32) ! [1,1,0,0,...];The code “c” is deﬁned from the generator matrix “m” by using the command“c:= LinearCode(m);”. All codewords of weight 8 in “c” are obtained byusing the command “Words(c,8)”. The design Di is obtained by the com-mand “Design” and furthermore, the derived design D�i is obtained by thecommand “Contraction”. By the following program, the graph Gi is con-structed from the generator matrix of the code Ci as in the previous section.In the program, we denote G1, G2, G3, G4, G5 by G1, G2, G3, G4, G5, and thegenerator matrices of C1, C2, C3, C4, C5 by M1, M2, M3, M4, M5, respectively.SRGFromCode := function(m);c := LinearCode(m);d := Design<3,32 | {Support(w) : w in Words(c,8) }>;d2 := Contraction(d,Point(d,32));b := BlockSet(d2);v := #b;e := {};for i in [1..v-1] dofor j in [i+1..v] doif #(b.i meet b.j) eq 3 thene := e join {{i,j}};end if;end for;end for;g := Graph<v | e>;return g;end function;G1 := SRGFromCode(M1);G2 := SRGFromCode(M2);G3 := SRGFromCode(M3);3Naoyuki HORIGUCHI, Hiroyuki NAKASORA, and Takehisa WAKABAYASHIthe number of common neighbors of x and y is k, λ or µ according as x andy are equal, adjacent or non-adjacent respectively.Let D = (X,B) be a quasi-symmetric design with the intersection num-bers λ1, λ2 (λ1 < λ2). The block graph of D is a graph whose vertices are theblocks of D and whose edges are the pairs (Y, Z) of blocks with |Y ∩Z| = λ2.Then it is known that the block graph is strongly regular (cf. [2, Theorem5.3], [5, Theorem 37.7]).Tonchev [8] showed that there are exactly ﬁve non-isomorphic quasi-symmetric 2-(31, 7, 7) designs. Five strongly regular graphs with parame-ters (155, 42, 17, 9) are obtained from the ﬁve designs. Then Stoichev [7]proved that the ﬁve graphs are non-isomorphic by comparing the automor-phism groups. On the other hand, strongly regular graphs with the sameparameters are also obtained from the Steiner triple systems STS(31). It isknown that if v ≥ 15, non-isomorphic Steiner triple systems STS(v) lead tonon-isomorphic strongly regular graphs [1]. In this paper, we consider thecliques of maximum size in the ﬁve graphs obtained from the quasi-symmetricdesigns by using Magma. We will give an alternative proof of Theorem inStoichev [7], and further we will prove the following theorem:Theorem 1. There exist no isomorphisms among the block graphs from thequasi-symmetric 2-(31, 7, 7) designs and the Steiner triple systems STS(31)except the isomorphism between the block graphs of the point-plane designand the point-line design of PG(4, 2).2 Quasi-symmetric designs and strongly reg-ular graphsWe begin this section with Tonchev’s result [8]. Conway and Pless [3]showed that there exist exactly ﬁve inequivalent binary extremal doubly-even self-dual [32, 16, 8] codes. We denote these codes by C1, C2, C3, C4 andC5, whose components are q32, r32, 2g16, 8f4 and 16f2, respectively [3, TableIII]. The codewords of weight 8 in Ci form a 3-(32, 8, 7) design Di by theAssmus–Mattson theorem [2, Theorem 14.11]. As a derived design of Di, aquasi-symmetric 2-(31, 7, 7) design D�i is obtained. Its intersection numbersare 1 and 3. In addition, all derived designs of Di are isomorphic since theautomorphism group of Di acts transitively on the points. It is shown in [8]that the ﬁve quasi-symmetric 2-(31, 7, 7) design D�i (i = 1, 2, . . . , 5) obtainedfrom Di are non-isomorphic.Let Gi be the block graph of D�i (i = 1, 2, . . . , 5). The following theoremis proved by Stoichev [7].2On the strongly regular graphs obtained from quasi-symmetric 2-(31, 7, 7) designsTheorem 2. The ﬁve graphs G1, G2, G3, G4, G5 are non-isomorphic.We will give an alternative proof of the theorem. The generator matrixof the code Ci is obtained from Sloane [6]. We set the generator matrix inMagma as follows:m := KMatrixSpace(GF(2),16,32) ! [1,1,0,0,...];The code “c” is deﬁned from the generator matrix “m” by using the command“c:= LinearCode(m);”. All codewords of weight 8 in “c” are obtained byusing the command “Words(c,8)”. The design Di is obtained by the com-mand “Design” and furthermore, the derived design D�i is obtained by thecommand “Contraction”. By the following program, the graph Gi is con-structed from the generator matrix of the code Ci as in the previous section.In the program, we denote G1, G2, G3, G4, G5 by G1, G2, G3, G4, G5, and thegenerator matrices of C1, C2, C3, C4, C5 by M1, M2, M3, M4, M5, respectively.SRGFromCode := function(m);c := LinearCode(m);d := Design<3,32 | {Support(w) : w in Words(c,8) }>;d2 := Contraction(d,Point(d,32));b := BlockSet(d2);v := #b;e := {};for i in [1..v-1] dofor j in [i+1..v] doif #(b.i meet b.j) eq 3 thene := e join {{i,j}};end if;end for;end for;g := Graph<v | e>;return g;end function;G1 := SRGFromCode(M1);G2 := SRGFromCode(M2);G3 := SRGFromCode(M3);3Naoyuki HORIGUCHI, Hiroyuki NAKASORA, and Takehisa WAKABAYASHIG4 := SRGFromCode(M4);G5 := SRGFromCode(M5);Table 1. Resultsgraph maximum size of cliques � of cliquesG1 7 2015G2 15 31G3 15 1G4 10 21G5 10 2We will consider the sizes and the numbers of maximum cliques of G1,G2, G3, G4, G5. Here a clique is an induced complete subgraph. We canuse the command “MaximumClique” to compute the maximum size of thecliques of the graphs. Moreover we can compute the numbers of the cliquesof maximum size by using command “AllCliques(G,n)”, where G is a graphand n is a size of clique. For example,> #MaximumClique(G1);7> #AllCliques(G1,7);2015The sizes and numbers of maximum cliques of graphs are invariants underisomorphisms of graphs. So the results in Table 1 prove Theorem 2.3 Steiner triple systems and strongly regulargraphsA 2-(v, 3, 1) design D is called a Steiner triple system and is denoted bySTS(v). It is known that there is an STS(v) if and only if v ≡ 1, 3 (mod 6).The total number of blocks is equal to v(v − 1)/6 and the number of blockscontaining a point is equal to (v− 1)/2. Also, D is a quasi-symmetric designwith the intersection numbers 0 and 1. It follows from [5, Theorem 37.7]that the block graph of an STS(v) (v > 7) is a strongly regular graph withparameters (v(v−1)6 ,3v−92 ,v+32 , 9). Hence the block graph of an STS(31) hasparameters (155, 42, 17, 9) which are equal to those of the strongly regulargraph obtained from a quasi-symmetric 2-(31, 7, 7) design. In general, thefollowing theorem is known [1].4Naoyuki HORIGUCHI, Hiroyuki NAKASORA, and Takehisa WAKABAYASHIG4 := SRGFromCode(M4);G5 := SRGFromCode(M5);Table 1. Resultsgraph maximum size of cliques � of cliquesG1 7 2015G2 15 31G3 15 1G4 10 21G5 10 2We will consider the sizes and the numbers of maximum cliques of G1,G2, G3, G4, G5. Here a clique is an induced complete subgraph. We canuse the command “MaximumClique” to compute the maximum size of thecliques of the graphs. Moreover we can compute the numbers of the cliquesof maximum size by using command “AllCliques(G,n)”, where G is a graphand n is a size of clique. For example,> #MaximumClique(G1);7> #AllCliques(G1,7);2015The sizes and numbers of maximum cliques of graphs are invariants underisomorphisms of graphs. So the results in Table 1 prove Theorem 2.3 Steiner triple systems and strongly regulargraphsA 2-(v, 3, 1) design D is called a Steiner triple system and is denoted bySTS(v). It is known that there is an STS(v) if and only if v ≡ 1, 3 (mod 6).The total number of blocks is equal to v(v − 1)/6 and the number of blockscontaining a point is equal to (v− 1)/2. Also, D is a quasi-symmetric designwith the intersection numbers 0 and 1. It follows from [5, Theorem 37.7]that the block graph of an STS(v) (v > 7) is a strongly regular graph withparameters ( v(v−1)6 ,3v−92 ,v+32 , 9). Hence the block graph of an STS(31) hasparameters (155, 42, 17, 9) which are equal to those of the strongly regulargraph obtained from a quasi-symmetric 2-(31, 7, 7) design. In general, thefollowing theorem is known [1].4On the strongly regular graphs obtained from quasi-symmetric 2-(31, 7, 7) designsTheorem 3. Let D (resp. D�) be an STS(v) and Γ (resp. Γ�) be a stronglyregular graph with parameters (v(v−1)6 ,3v−92 ,v+32 , 9) obtained from D (resp.D�) as the block graph.(1) If v ≥ 19, then Γ has exactly v maximal cliques of maximum size v−12 .(2) If v ≥ 15 and D is non-isomorphic to D�, then Γ is non-isomorphicto Γ�.By Theorem 3(1), if v = 31, the number of the maximal cliques of maxi-mum size of Γ is equal to that of the graph G2 in Table 1. The four graphsG1, G3, G4, G5 can not be obtained from an STS(31). We will give an ex-ample of STS(31) whose block graph is isomorphic to G2. We note thatthere are at least 6 × 1016 non-isomorphic STS(31) (cf. [4]). Let V be an(n+1)-dimensional vector space over the q-element ﬁeld Fq. We consider theprojective geometry PG(n, q) consisting of the set of all vector subspaces ofV . In the case n = 4 and q = 2, the points and planes of PG(4, 2) form aquasi-symmetric 2-(31, 7, 7) design and the points and lines of PG(4, 2) forma Steiner triple system STS(31). Here, the following proposition is known[2, Exercise 5 of Chapter 5].Proposition 4. For n > 2, the points and (n− 2)-ﬂats of PG(n, q) form aquasi-symmetric design, whose block graph is isomorphic to the block graphof the point-line design of PG(n, q).By this proposition, the block graph G2 of the quasi-symmetric 2-(31, 7, 7)design D�2 is isomorphic to the block graph of the point-line design STS(31)of PG(4, 2). By Theorem 3(2), this is the only one isomorphism among theblock graphs of the quasi-symmetric 2-(31, 7, 7) designs and the block graphsof Steiner triple systems STS(31).AcknowledgmentsThe authors would like to thank Masaaki Harada, the supervisor of thethird author, and Masaaki Kitazume, the supervisor of the ﬁrst and secondauthors, for useful discussions. The authors would also like to thank VladimirTonchev for providing the reference [7].References[1] L. Babai, Automorphism groups, isomorphism, reconstruction, in Hand-book of combinatorics, Vol. 1, 2, R. L. Graham, M. Gro¨tschel and L.Lova´sdz (Eds.), Elsevier, Amsterdam, 1995, 1447–1540.5Naoyuki HORIGUCHI, Hiroyuki NAKASORA, and Takehisa WAKABAYASHI[2] P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and theirLinks, London Mathematical Society Student Text 22, Cambridge Uni-versity Press, Cambridge, 1991.[3] J. H. Conway and V. Pless, On the enumeration of self-dualcodes,J. Combinatorial Theory Ser. A 28 (1980), 26–53.[4] R. Mathon and A. Rosa, 2-(v, k, λ) designs of small order, in CRC Hand-book of Combinatorial Designs, C.J. Colbourn and J.H. Dinitz, (Eds.),CRC Press, Boca Raton, 1996, 3–41.[5] M. S. Shrikhande, Quasi-symmetric designs, in CRC Handbook of Com-binatorial Designs, C.J. Colbourn and J.H. Dinitz, (Eds.), CRC Press,Boca Raton, 1996, 430–436.[6] N. J. A. Sloane, “The 85 doubly-even self-dual [32, 16, 8] binary codeof length 32”, in A Library of Linear (and Nonlinear) Codes, publishedelectronically at http://www.reseach.att.com/~njas/codes/.[7] S. D. Stoichev, The nonisomorphism of the strongly regular graphs de-rived from the quasi-symmetric 2-(31, 7, 7) designs. C. R. Acad. BulgareSci. 40 (1987), 33–35.[8] V. D. Tonchev, Quasi-symmetric 2-(31, 7, 7) designs and a revision ofHamada’s conjecture, J. Combinatorial Theory Ser. A 42 (1986), 104–110.6Bull. of Yamagata Univ., Nat. Sci., Vol.16, No.1, Feb. 2005粒子ビームを用いたシャワー型検出器の性能評価吉田浩司∗（平成 16年 7月 15日 受理）AbstractThe performance of a lead/scintillator sandwich-type detector witha wave length shifter ﬁber readout has been tested by using 0.4–2.0GeV/c electrons and pions. It has been fabricated as a prototypecounter for the KEK E391a experiment. For electrons the prototypecounter provides a very linear response to the incident energy. Theenergy resolution is obtained to be σ/E (%) = 5.10± 0.03/�E (GeV)⊕ 0.00 ± 0.26. A light yield of 5.75± 0.03 photoelectrons/MeV isobtained for the incident energy of γ rays. It corresponds to a lightyield of 19.6± 0.1 photoelectrons/MeV for the energy deposit in aplastic scitillator. These values ensure the detection ineﬃciency of10−4 for photon veto counters of the E391a experiment.1 序論KEK E391a実験のために，Wave Length Shifter (WLS) Fiberを集光系に採用したサンドイッチ型電磁シャワーカロリメーターを試作し，π，eなどの粒子ビームを用いて性能評価実験をおこなった。本論文ではその実験結果について報告する。KEK E391a実験は，K0L → π0νν¯ 反応の存在を探索し，K崩壊におけるその反応の分岐比を測定しようとするものである [1][2]。標準理論により予想されている分岐比は 10−11程度と大変小さいものであるが [3][4]，一方で理論計算による不定性も数%程度と大変小さいと考えられている [5][6]。したがって，この反応の分岐比測定は，CKM行列要素の ImVtdに関して大変クリー∗Department of Physics, Faculty of Science, Yamagata University, Yamagata990–8560, Japan. (Phone: +81-23-628-4570; Fax: +81-23-628-4567; E-mail:yoshida@sci.kj.yamagata-u.ac.jp)7

On the strongly regular graphs obtained from quasi-symmetric 2-(31,7,7) designs

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On the strongly regular graphs obtained from quasi-symmetric 2-(31,7,7) designs

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