Automorphism groups of bipartite Kneser type-k graphs

For integers n > 1, k >= 1, let I-n = {1, 2, 3, ..., n}. Let phi(I-n) be the set of all non-empty subsets of I-n. Let V-1 be the set of k-element subsets of I-n, and V-2 = phi(I-n) - V-1. A bipartite Kneser type-k graph H-T(n, k) is defined with parts V-1 and V-2, and a vertex A is an element of V-1 is adjacent to a vertex B is an element of V-2 if and only if A subset of B or B subset of A. The algebraic properties of bipartite Kneser type-k graphs are investigated. For any integers n >= 3, the automorphism groups of bipartite Kneser type-1 graphs are isomorphic to the symmetric group S-n. The bipartite Kneser type-1 graph's Wiener index, peripheral Wiener index, and peripheral Hosoya polynomial were established. For integers n and k, n > 2 and 2k < n or 2k > n, 1 <= k < n, the automorphism group of H-T(n, k) is isomorphic to the symmetric group S-n. For n is even and 2k = n, the automorphism group of H-T(n, k) is isomorphic to S-n x Z(2), where Z(2) is the cyclic group of order 2.