Distance-biregular graphs with 2-valent vertices and distance-regular line graphs

Abstract

A graph G is called distance-regularized if each vertex of G admits an intersection array. It is known that every distance-regularized graph is either distance-regular (DR) or distance-biregular (DBR). Note that DBR means that the graph is bipartite and the vertices in the same color class have the same intersection array. A (k, g)-graph is a k-regular graph with girth g and with the minimum possible number of vertices consistent with these properties. Biggs proved that, if the line graph L( G) is distance-transitive, then G is either K 1, n or a (k, g)-graph. This result is generalized to DR graphs by showing that the following are equivalent: (1) L( G) is DR and G ≠ K 1, n for n ≥ 2, (2) G and L( G) are both DR, (3) subdivision graph S( G) is DBR, and (4) G is a (k, g)-graph. This result is used to show that a graph S is a DBR graph with 2-valent vertices iff S = K 2,′ or S is the subdivision graph of a (k, g)-graph. Let G (2) be the graph with vertex set that of G and two vertices adjacent if at distance two in G. It is shown that for a DBR graph G, G (2) is two DR graphs. It is proved that a DR graph H without triangles can be obtained as a component of G (2) if and only if it is a (k, g)-graph with g ≥ 4.