A graph is walk-regular if the number of cycles of length rooted at a given vertex is
a constant through all the vertices. For a walk-regular graph G with d+1 different
eigenvalues and spectrally maximum diameter D = d, we study the geometry of its
d-cliques, that is, the sets of vertices which are mutually at distance d. When these
vertices are projected onto an eigenspace of its adjacency matrix, we show that they
form a regular tetrahedron and we compute its parameters. Moreover, the result...
A graph is walk-regular if the number of cycles of length rooted at a given vertex is
a constant through all the vertices. For a walk-regular graph G with d+1 different
eigenvalues and spectrally maximum diameter D = d, we study the geometry of its
d-cliques, that is, the sets of vertices which are mutually at distance d. When these
vertices are projected onto an eigenspace of its adjacency matrix, we show that they
form a regular tetrahedron and we compute its parameters. Moreover, the results
are generalized to the case of k-walk-regular graphs, a family which includes both
walk-regular and distance-regular graphs, and their t-cliques or vertices at distance
t from each other.
Citation
Dalfo, C.; Fiol, M. A.; Garriga, E. On t-cliques in k-walk-regular graphs. "Electronic notes in discrete mathematics", Agost 2009, vol. 34, p. 579-584.