武元, 英夫

高村, 正樹

Kyoto University Research Information Repository

TitleTHE NUMERICAL RADIUS OF AN INFINITE DIRECTEDREGULAR GRAPH(Recent topics on the operator theoryabout the structure of operators)Author(s)高村, 正樹; 武元, 英夫Citation数理解析研究所講究録 (1997), 979: 99-103Issue Date1997-02URL http://hdl.handle.net/2433/60850RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityTHE NUMERICAL RADIUS OF AN INFINITEDIRECTED REGULAR GRAPH$G=(V, E)$ , , $V$ , $E$$e=(u, v)\in E$ , $u$ $e$ , $v$ $e$ , $v$$d^{-}(v)$ , $v$$d^{+}(v)$ , , $G$bounded valency, ,, Mohar [4], [3]$e_{v}(u)=\delta_{v,u}$ $\{e_{v} ; v\in V\}$$H=\ell^{2}(V)$ , $G=(V, E)$ $A=A(G)$;$\mathrm{D}\mathrm{o}\mathrm{m}(A)=\{x=\sum_{v\in V}xve_{v}\in H$ ; $\sum_{u\in V}|\sum_{v\in D^{-}(u)}X_{v}|^{2}<+\infty\}$,$Ax= \sum$ $\sum$ $x_{v}e_{u}$ .$u\in Vv\in D-(u)$$D^{-}(u)$ , $u$ , $A=$$A(G)$ $G$ $A(G)$ , $G$ bounded valency, $G$ bounded valency ,$G$ $A(G)$ , $\sigma(G)$$r’(G)$ , $w(G)$$r(G)= \sup\{|\lambda| ; \lambda\in\sigma(G)\}$ , $w(G)= \sup\{|\langle A(G)x, X\rangle| ; ||x||=1, x\in H\}$,$r(G)\leqq w(G)\leqq||A(G)||\leqq\sqrt{d^{+}d^{-}}$979 1997 99-103 99$d^{+}= \max_{v\in V}d+(v),$ $d^{-}= \max_{v\in V}d^{-}(v)$ $G$ 1 ,$A(G)$ , $d$ $G$ ,$r(G)=w(G)=||A(G)||\leqq d$, $k$ , k-$G$ , $k$- $r(G)=k$, ,1([5]). $G=T_{3}$ (homogeneous tree);, $G$ 3- , $r(G)=2\sqrt{2}$, Biggs [1] , k-$G$ , $0$ , $r(G)=k$, [2] , k-$G$ $0$ $A(G)$ normaloid , $r(G)=k$, $G=(V, E)$$2([2])$ . $G=(V, E)$ $V$ X , $X$$V\backslash X$ $\partial^{+}X$ , $V\backslash X$ $X$ $\partial^{-}X$,$i^{\pm}(G)= \inf\{\frac{|\partial^{\pm}X|}{|X|}\cdot X$ $V\text{ }\}$100,$i(G)= \max\{i^{+}(G), i^{-}(G)\}$$i(G)$ $G$ , $|\cdot|$ ,3 $([2])$ . $k$- $G=(V, E)$ , $i(G)=0$ ,$A(G)$ normaloid , $r(G)=k$, $i_{0}(G)$ , $i_{0}(G)=0$, $r(G)=w(G)=||A(G)||=k$4 $G=(V, E)$ ,$i_{0}(G)= \inf\{\frac{|\partial^{+}X|+|\partial^{-}X|}{2|X|}$ ; $X$ $V$ $\text{ _{}\mathrm{r}}\mathrm{J}*\mathrm{f}\mathrm{l}^{-\text{ }\mathrm{I}}$$i_{0}(G)$ $G$5 $G=(V, E)$ , $uv\in E$ , $(u, v)\in E$$(v, u)\in E$ , $i(G)$ $i_{0}(G)$ , [1]–$G=(V, E)$ , $x\in\ell^{2}(V)$ , $\triangle(x)$$\triangle(x)=\sum_{v(u,)\in E}|_{X_{u}^{2}}-x_{v}|2$., $V$ $X$ , $x=\{x_{v}\}\in l^{2}(X)$ , $v\in X$$\tilde{x}_{v}=x_{v},$ $v\in V\backslash X$ $\tilde{x}_{v}=0$ , $\tilde{x}=\{\overline{x}_{v}\}\in\ell^{2}(V)$$\text{ }$ $6$ $G=(V, E)$ $V$ $X$ ,$x\in\ell^{2}(X)$$\triangle(\tilde{x})\geqq 2i\mathrm{o}(G)||_{\tilde{X}}||^{2}$ .,7 $k$- $G$ ,$\frac{(i_{0}(G))2}{2k}\leqq k-w(G)\leqq i_{0}(G)$101,8.$k$- $G=(V, E)$ , $i_{0}(G)=^{\mathrm{o}}$ , $A(G)$ normaloid, $r(G)=k$ , $k=r(G)=w(G)=||A(G)||$ ,$w(G)=k$ $i_{0}(G)=0$ , $r(G)=w(G)=||A(G)||=k$, $k$ $G$ , $i_{0}(G)=^{\mathrm{o}}$ $r(G)=w(G)=$$||A(G)||=k$ , 2 $i(G)$$i_{0}(G)$, $i(G)\leqq 2i_{0}(G)$ , $i_{0}(G)=0$$i(G)=0$ ,9 $G$$\nearrow^{\bullet}$. . .$\bullet-.\cdotarrow.\backslash \nearrow^{\bulletarrow\bullet}\bulletarrow\bullet\cdot$.$.\cdot\cdot$.$\backslash \bullet\cdots$1, $i(G)=0$ , $i_{0}(G)=2$, $G=(V, E)$ k- , 3 , $i(G)=0$$r(G)=w(G)=||A(G)||=k$ , 8 $i_{0}(G)=0$, $C_{J}^{\gamma}$ , $i(G)=0$ $i_{0}(c)=0$, [6] elementary ratio ,10 ([6]) $G=(V, E)$ ,$\epsilon(G)=\sup\{\frac{|E(X)|}{|X|}$ ; $G’=(X, E(X))$ 1 $G\text{ }\}$$G$ elementary ratio11. $([6])$ $k$- $G$ ,(1) $i(G)=0$ $\epsilon(G)=k$(2) $\epsilon(G)=k$ , $A(G)$ normaloid $r(G)=k$102, 8 11 ,12. $k$- $G=(V, E)$ , 4(1) $r(G)=w(G)=||A(G)||=k$(2) $i(G)=0$(3) $i_{0}(G)=0$(4) $\epsilon(G)=k$[1] N. L. Biggs, B. Mohar $\mathrm{a}\mathrm{I}\mathrm{u}\mathrm{d}$ J. Shawe-Taylor, The spectral radius of infinite graphs,Bull. London Math. Soc., 20 (1988), 116-120.[2] $\mathrm{J}.\mathrm{I}$ . Fujii, M. Fujii, H. Sasaoka and Y. Watatani, The Spectrum of an infinitedirected graph, Math. Japonica, 36 (1991), 607-625.[3] M. Fujii, H. Sasaoka and Y. Watatani, Adjacency operators of infinite directedgraphs, Math. Japonica, 34 (1989), 727-735.[4] B. Mohar, The spectrum of an infinite graph, Linear Algebra Appl. 48 (1982),245-256.[5] B. Mohar and W. Woess, A survey on spectra of infinite graphs, Bull. LondonMath. Soc., 21 (1989), 209-234.[6] Y. Seo, The spectral radius of an infinite directed graph, Math, Japonica, 39 (1994),373-379103

THE NUMERICAL RADIUS OF AN INFINITE DIRECTED REGULAR GRAPH(Recent topics on the operator theory about the structure of operators)

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THE NUMERICAL RADIUS OF AN INFINITE DIRECTED REGULAR GRAPH(Recent topics on the operator theory about the structure of operators)

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