By Ulrich Knauer
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Extra resources for Algebraic Graph Theory: Morphisms, Monoids and Matrices
R/ (b) dij is the smallest r 2 N with aij > 0 and r < n, if such an r exists; (c) dij D 1 otherwise. r/ (b) rij D 1 if and only if there exists r < n with aij > 0; (c) rij D 0 otherwise. 8. G//r for r Ä n and for all i . 4 Chapter 2 Graphs and matrices Endomorphisms and commuting graphs We brieﬂy discuss two aspects of the adjacency matrix which have not gained much attention so far. 1. e. a mapping of the set into itself. j /Di ej , where ej is the j th row of the identity matrix In and 0 is the row of zeros with n elements.
G/ P 1 ; where P is an n n row permutation matrix which comes from the identity matrix In upon performing row permutations corresponding to f . Proof. e. that G 0 comes from G by permutation of the vertices. G/, rows and columns are permuted correspondingly. G/ P 1 , where P is the corresponding row permutation matrix. Left multiplication by P then permutes the rows and right multiplication by P 1 permutes the columns. G/ P 1 where P is a permutation matrix. Then there exists a mapping f W V ! e.
7. Let G be a graph with n vertices. r/ (b) dij is the smallest r 2 N with aij > 0 and r < n, if such an r exists; (c) dij D 1 otherwise. r/ (b) rij D 1 if and only if there exists r < n with aij > 0; (c) rij D 0 otherwise. 8. G//r for r Ä n and for all i . 4 Chapter 2 Graphs and matrices Endomorphisms and commuting graphs We brieﬂy discuss two aspects of the adjacency matrix which have not gained much attention so far. 1. e. a mapping of the set into itself. j /Di ej , where ej is the j th row of the identity matrix In and 0 is the row of zeros with n elements.