篠原, 雅史

野崎, 寛

Kyoto University Research Information Repository

Title距離集合における点の配置問題とグラフ (デザイン、符号、グラフおよびその周辺)Author(s)野崎, 寛; 篠原, 雅史Citation数理解析研究所講究録 (2013), 1844: 39-49Issue Date2013-07URL http://hdl.handle.net/2433/195030RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityHiroshi NozakiDepartment of Mathematics,Aichi University of Education*Suzuka National College of Technology12 distance set1 “ ”1 “ ( ) ”$P$-$k$ ( ) k-distance set([1], pp284) ( ) optimaldistance setdistance setdistance set4k-distance set $k$-diameter graph orthogonal graph32 distance set$\bullet$$\mathbb{R}^{d_{:}}d$$\bullet$$S^{d-1}:\mathbb{R}^{d}$$\bullet$ $X\subset S^{d-1}$ $X=-X$ $X$ antipodal*Research supported by JSPS KAKENHI Grant Number 24740076.1844 2013 39-49 39$\bullet$ FOr $P=(p_{1},p_{2}, \ldots,p_{d}),$ $Q=(q_{1}, q_{2}, \ldots, q_{d})\in \mathbb{R}^{d},$$PQ=\sqrt{\sum_{i=1}^{d}(p_{i}-q_{i})^{2}}$Definition 2.1.$\bullet$ For $X\subset \mathbb{R}^{d}(|X|<\infty),$ $A(X);=\{PQ:P, Q\in X, P\neq Q\},$ $a(X)=$$|A(X)|.$$\bullet$ $a(X)=k$ $X$ k-distance set$\bullet$ k-distance set k-distance set( )$X_{1} X_{2} X_{3} X_{4}$$A(X_{1})=\{1\}$ $A(X_{2})=\{1, \sqrt{2}\}$ $A(X_{3})=\{1, \tau\}A(X_{4})=\{1, \sqrt{2}, \sqrt{3}\}$1-distance set 2-distance set 2-distance set 3-distance set$( \tau=\frac{1+\sqrt{5}}{2})$$n$ $n$ $\lfloor\frac{n}{2}\rfloor$ -distance set, 1220 3-distance set, 5-distance set$k$-distance set $X\subset \mathbb{R}^{d}$ $d$ $a_{-}(X)$distance setDefinition 2.3.$\bullet$ $DS(d, k)$ $:= \max${ $|X|$ : $X$ is a $k$ -distance set in $\mathbb{R}^{d}$ }.$\bullet$$\mathbb{R}^{d}$ k-distance set $X$ $|X|=DS(d, k)$ optimalProblem 2.4. $k,$ $d$$\bullet$ $DS(d, k)$$\bullet$$\mathbb{R}^{d}$ $k$-distance set $DS(d, k)$402.1 Tight two-distance setsTheorem 2.5. $S^{d-1}\subset \mathbb{R}^{d}$ ($re\mathcal{S}P$ . antipodal) $k$ -distance set$(\begin{array}{ll}d+k -1k \end{array})+(\begin{array}{ll}d+k -2k -1\end{array}) (resp. 2(\begin{array}{ll}d+k -2k -1\end{array}))$tight (antipodal) $k$-distance set tight k-distance setProblem 2.6. $d=2,6,22$ $S^{d-1}$ tight 2-distance set$d\geq 3$ $d=(2n-1)^{2}-3$ ( $n$ : )(Bannai-Damerell [2, ])Remark 2.7. tight 2-distance set tighttight 2-distance settight antipoda13-distance set5 $2O$tight 2-distance set$\bullet$$S^{d-1}$ tight 2-distance set$\Leftrightarrow S^{d}$ tight antipoda13-distance set$\bullet$$S^{1}$ : tight 2-distance set ( 5 )$\Leftrightarrow S^{2}$ :tight antipoda13-distance set ( 20 )133.1Definition 3.1. $G=(V(G), E(G))$41$\bullet$ $Y\subset V(G)$$Y$ $G$ $rightarrow v\sim w$ for $_{v,w}\in Y$$\bullet$ $\alpha(G):=\max$ { $|Y|$ : $Y$ is an $ind$. set in $G$} : $G$Definition 3.2. (Diameter graph )$\bullet$ $D(X):= \max A(X):X\subset \mathbb{R}^{d}$ diameter$\bullet$ $G:=DG(X):X$ diameter graph$\{\begin{array}{ll}V(G)=X, For P, Q\in X, P\sim Q if PQ=D(X) .\end{array}$Example 3.3. $P_{n}$ $n$ $C_{n}$ $n$$n$$DG(R_{2m+1})=C_{2m+1}, DG(R_{2m})=m\cdot P_{2}.$diameter graph $X$ k-distance set$X$ diameter graph $G=DG(X)$ $V(G)$$G$ $H$ $Y\subset X$ $X$$D(X)$ $(k-1)$-distance set$X \supset Y$$\uparrow$ $\downarrow$$V(DG(X))$ $\supset$ $H$ : an independent set$Y$ $(k-1)$-distance set.Lemma 3.4. $\{A, B, P, Q\}\in X\subset \mathbb{R}^{2}$ $AB=PQ=D(X)$2 $AB,$ $PQ$diameter graphProposition 3.5. $X\subset \mathbb{R}^{2}$ $G=DG(X)$(i) $G$ $C_{2m}(m\geq 2)$ ;(ii) $G$ $C_{2m+1}$ $V(G)\backslash V(C_{2m+1})$$G$Proposition 3.6. $X\subset \mathbb{R}^{2}(|X|=n)$ $G=DG(X)$ diametergraph $G\neq C_{n}$ $\alpha(G)\geq\lceil\frac{n}{2}\rceil$Example 3.7. $X$ 9-point 4-distance set diameter graph$G=DG(X)$$\bullet$ $G=C_{9}$ $X=R_{9}$ ( $\cdot.\cdot X$ -distance set)$\bullet$ $G\neq C_{9}$ $\alpha(G)\geq\lceil\frac{9}{2}\rceil=5$ . $X$ 5-point (at most)3-distance set423.2 4 2-distance set22 2-distance set $\mathbb{R}^{2}$$\mathbb{R}^{3}$$P$2-distance set23$K_{4}$ 2 5$G_{1} G_{2} G_{3} G_{4} G_{5}$( 2 )$\bullet$ 2-distance set$\bullet$ 2 $G$ $\overline{G}$$\bullet$ $X$433.3Definition 3.8. $G$ orthogonal graph$H\subset V(G)(2\leq|H|\leq|V(G)|-2)$$N(w_{1})\backslash H=N(w_{2})\backslash H (^{\forall}w_{1}, w_{2}\in H)$$N(w)=\{v\in V(G)|vw\in E(G)\}.$Definition 3.9. $A(X)=\{a, b\}(a<b)$ 2-distance set $X$( ) $G$$\{\begin{array}{l}V(G)=X,(p, q)\in E\Leftrightarrow d(p, q)=a.\end{array}$2-distance set 22-distance setExample 3.10. $G$ orthogonal graph2-distance set $X$$G\Leftrightarrow X\subset \mathbb{R}^{4}$$\bullet$ $Y$ spherical $Y\perp X\backslash Y$$\bullet$ $\dim(Y)+\dim(X\backslash Y)=4$ $Y\subset \mathbb{R}^{2},$ $X\backslash Y\subset \mathbb{R}^{2}$$\bullet$$Y\subset \mathbb{R}^{2}$$\bullet$ $X\backslash Y$ 4$t$$($$xy$ -plane $zw$ -plane44$a=\sqrt{\frac{7-4\tau}{20}},$ $b= \sqrt{\frac{11\tau+7}{20}}(\tau=\frac{1+\sqrt{5}}{2})$ .4 2-distance set4.1 2-distance setTheorem 4.1 (Einhorn-Schoenberg [4]). $n$ $n-1$2-distance set $n-2$12-distance set / $N$ 2-distance set $M(G)$$d(G)$Example 4.2. 4 24.2 Tight graphTheorem 4.3. $G$ $n$$d(G)+d(\overline{G})\geq n-1.$Definition 4.4. tight graphProblem 4.5. Tight graphExample 4.6. tight graph1. ( : $r$ $K_{i,i,\ldots,i}$ ).2. $K_{i,j}$ ($i=j$ ).3. 9 $G_{1},$ $G_{2},$ $G_{3}.$$(d(G_{i})=4, G_{1}=\overline{G_{1}}, G_{2}=\overline{G_{3}})$4. 45 $G$ . (Lisonek[5])Remark 4.7. 3 2-distance set non-sphericalnon-spherical 2-distance set $G_{1}$Example 3. 10451 2-distance setExample 4.8. $G=K_{nnn_{r}}1,2,\ldots,(n_{1}\geq n_{2}\geq\cdots\geq n_{r})$ $X=M(G)$(i) $n_{1}=n_{2}=\cdots=n_{t}>n_{t+1}$ $X$ spherical$d(G)=n-t, X\subset M(G’)$ .$G’=K_{n_{1},n_{1,\ldots,1}}n$ ( $r$ ).(ii) $n_{1}\neq n_{2}$ $X$ non-spherical $d(G)=n-2$4.3 Design and distance setTheorem 4.9 (Nozaki-$S$ . [9]). $d(G)+d(\overline{G})=n-1\Leftrightarrow X=M(G)$constant weight Euclidean 2-design 2-distance setstrongly regular graph $\Rightarrow$ $??$?spherical 2$\Uparrow$-design $\Rightarrow$ constant weight$Euclidean\Downarrow$$2$-designand 2-distance set and 2-distance sets$d(G)+d() \frac{\Downarrow}{G}=n-1$ $d(G)+d( \frac{\Downarrow}{G})=n-1$and sphericalConjecture 4.10. $\mathbb{R}^{3}$ 5-distance set 20$DS(3,5)=20.$spherical designTheorem 4.11 (Nozaki-Suda [10]). $X$ antipodal 5-distance set $\max-$imum strength 5 (sphe $cal$ 5-design 6-design )$|X|\leq 20.$4.4 2-distance set2-distance setRoy ([11]) 5 2-distance setTheorem 4.12. $G(|V(G)|=n)$ 2-distance set $X=M(G)$ ,$A(X)=\{c, 1\}(c<1)$1. $c=(\tau_{1}+1)/\tau_{1}$ $d=n-m_{1}-1.$462 $c=(\tau_{1}+1)/\tau_{1}$ $d=n-m_{1}.$3. $c=(\tau_{2}+1)/\tau_{2}$ $d=n-m_{2}-2.$4. $c=(\tau_{2}+1)/\tau_{2}$ $d=n-m_{2}-1.$5, $c<(\tau_{1}+1)/\tau_{1},$ $c\neq(\tau_{2}+1)/\tau_{2}$ $d=n-2.$$A(X)=\{c, 1\},$ $\tau_{i}$ $G$ $i$$m_{i}$[9] 5 Type 1,2,5 2-distanceset spherical Type 3 $G$non-spherical $d(G)<n-2$Table A$d=4,5,6$Maximum cardinalities of two-distance sets for $d\leq 8$Table ATyPe 3 $d=4,5,6$Proposition 4.13.$\bullet$ $d=4:9$ non-spherical 2-distance set 3$\bullet$ $d=5:10$ non-spherical 2-distance set$\bullet$ $d=6:17$ non-sphsrical 2-distance setTable $B$Problem 4.14. TyPe 3475 Appendix$G_{4}$ $x_{0}=2$ $\frac{1}{2},$ $y_{0}=\sqrt{t^{2}+\frac{1}{4}},$ $z_{0}=\sqrt{-4t^{2}+3t-\frac{1}{4}}$$a= \frac{3-\sqrt{5}}{8}(=\sin^{2}\frac{\pi}{10}),$ $b= \frac{3+\sqrt{5}}{8}(=\sin^{2}\frac{3}{10}\pi)$References[1] Ei. Bannai and Et. Bannai,1999.[2] Ei. Bannai, R. M. Damerell, Tight spherical designs. II. J. London Math.Soc. (2) 21 (1980), no. 1, 13-30.[3] P. Delsarte, J.M. Goethals, and J.J. Seidel, Spherical codes and designs,Geom. Dedicata 6 (1977), no. 3, 363-388.[4] S.J. Einhom and I.J. Schoenberg, On euclidean sets having only twodistances between points. I. II. Nederl. Akad. Wetensch. Proc. Ser. $A$$69=$ Indag. Math. 28 (1966), 479-488, 489-504.[5] P. Lison\v{e}k, New maximal two-distance sets, J. Combin. Theory, Ser. $A$77 (1997), 318-338.48[6] 0.$R$ . Musin, Spherical two-distance sets, J. Combin. Theory, Ser. $A$ 116(2009), 988-995.[7] A. Neumaier, Distance matrices, dimension, and conference graphs, Ned-$erl$. Akad. $Weten\mathcal{S}ch$ . Indag. Math. 43 (1981), 385-391.[8] H. Nozaki and M. Shinohara, On a generalization of distance sets, $J.$Combin. Theory, Ser. $A$ 117 (2010), 810-826.[9] H. Nozaki and M. Shinohara, $A$ geometrical characterization of stronglyregular graphs, Linear Algebm and its Applications 437 (2012) 2587-2600.[10] H. Nozaki and S. Suda, Bounds on $s$-distance sets with strength $t$ . SIAMJ. Discrete Math. 25 (2011), no. 4, 1699-1713.[11] A. Roy, Minimal Euclidean representation of graphs, Discrete math.310 (2010), 727-733.49

キョリ シュウゴウ ニオケル テン ノ ハイチ モンダイ ト グラフ デザイン フゴウ グラフ オヨビ ソノ シュウヘン

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キョリシュウゴウニオケルテンノハイチモンダイトグラフデザインフゴウグラフオヨビソノシュウヘン

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キョリ シュウゴウ ニオケル テン ノ ハイチ モンダイ ト グラフ デザイン フゴウ グラフ オヨビ ソノ シュウヘン

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Kyoto University Research Information Repository

キョリシュウゴウニオケルテンノハイチモンダイトグラフデザインフゴウグラフオヨビソノシュウヘン