By Boris Weisfeiler (auth.)
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Extra resources for On Construction and Identification of Graphs
I s d e n o t e d b y ~ < ~ t)L. We s h a l l s h o w b e l o w ( s e e L e m m a 2) t h a t C can be brought into 49 block-diagonal f o r m with diagonal blocks of the s a m e degree, shall write I~I = d(C) and call I~I C = ]E ~ I . W e r m th___~e de~ree of the n o r m a l subcell ~ . fact mentioned above also implies that ~i< kxifi is of the f o r m w h e r e every X. is the matrix of a cell and 1 non-zero entry equal to one at position the n o r m a l subcell is simply The r Zi=lhi ~D X i, h.
Proof. 4. Evident. Proposition. ~=gL(Z). Y, Z Let 0L, ~ , 0~ be ideals in the algebras 0-6= 0-~(Y ~ Z), ~ = c<,~,~, 0L(y), respectively, 43 complementary to the id~ls~, ~ , of L6. T h e n ~-~ = ~ e ~ (direct sum of algebras ). Proof. 5. ~ Remark. 6. (cf. L6). This assertion algebras Remark. are rather that the notions of direct sum for algebras and close. The direct sum of graphs is an analogue of the direct product of permutation groups. and W shows N a m e l y , let G respectively. izer rings ~ ( G , V ) Let X and and H Y be permutation groups acting on V be the stationary graphs of the central- and ~ ( H , W ) , respectively.
Latter a r e paths of the f o r m it is sufficient to conk, the s a m e paths (j',i ~) along the k' . ir k' ,1 niajk = n a j jkl' = nkaij " c8. Take cl with f = 0: s 0 ~sajkasi and use 0 j. i k k' aij = aj, i' " sider, together with the paths Proof. along an e d g e of colour 0) a n d then continue b y an e d g e of colour c7. Proof. 3s a ki = Es6j, snjaki = n ja ki T h e s e c o n d equality is p r o v e d Geometrically, all paths of the f o r m analogously. equality (j,k,i) w h o s e c8 has the following m e a n i n g .