Edge-distance-regularity is a concept recently introduced by the authors which is similar to that of distance-regularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A
is edge-distance-regular when it is distance-regular around each of its edges and with the same intersection numbers for any
edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ, and d...
Edge-distance-regularity is a concept recently introduced by the authors which is similar to that of distance-regularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A
is edge-distance-regular when it is distance-regular around each of its edges and with the same intersection numbers for any
edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ, and derive
some characterizations. In particular, it is shown that a graph is edge-distance-regular if and only if its k-incidence matrix is
a polynomial of degree k in A multiplied by the (standard) incidence matrix. Also, the analogue of the spectral excess theorem
for distance-regular graphs is proved, so giving a quasi-spectral characterization of edge-distance-regularity. Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.
Citation
Camara, M. [et al.]. Edge-distance-regular graphs. "Journal of combinatorial theory. Series A", 2011.