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On d-Walk Regular Graphs | | Ernesto Estrada
; Jose A. de la Pena
; | | Date: |
30 Mar 2013 | | Abstract: | Let G be a graph with set of vertices 1,...,n and adjacency matrix A of size
nxn. Let d(i,j)=d, we say that f_d:N->N is a d-function on G if for every pair
of vertices i,j and k>=d, we have a_ij^(k)=f_d(k). If this function f_d exists
on G we say that G is d-walk regular. We prove that G is d-walk regular if and
only if for every pair of vertices i,j at distance <=d and for d<=k<=n+d-1, we
have that a_ij^(k) is independent of the pair i,j. Equivalently, the single
condition exp(A)*A_d=cA_d holds for some constant c, where A_d is the adjacency
matrix of the d-distance graph and * denotes the Schur product. | | Source: | arXiv, 1304.0125 | | Services: | Forum | Review | PDF | Favorites | |
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