Automorphism groups of bipartite Kneser type-k graphs

Abstract

For integers [Formula: see text], [Formula: see text], let [Formula: see text]. Let [Formula: see text] be the set of all non-empty subsets of [Formula: see text]. Let [Formula: see text] be the set of [Formula: see text]-element subsets of [Formula: see text], and [Formula: see text]. A bipartite Kneser type-[Formula: see text] graph [Formula: see text] is defined with parts [Formula: see text] and [Formula: see text], and a vertex [Formula: see text] is adjacent to a vertex [Formula: see text] if and only if [Formula: see text] or [Formula: see text]. The algebraic properties of bipartite Kneser type-[Formula: see text] graphs are investigated. For any integers [Formula: see text], the automorphism groups of bipartite Kneser type-1 graphs are isomorphic to the symmetric group [Formula: see text]. The bipartite Kneser type-1 graph’s Wiener index, peripheral Wiener index, and peripheral Hosoya polynomial were established. For integers [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] or [Formula: see text], [Formula: see text], the automorphism group of [Formula: see text] is isomorphic to the symmetric group [Formula: see text]. For [Formula: see text] is even and [Formula: see text], the automorphism group of [Formula: see text] is isomorphic to [Formula: see text], where [Formula: see text] is the cyclic group of order 2.