Some algebraic properties of bipartite Kneser graphs

Abstract

Let n and k be integers with n > k ≥ 1 and [n] = {1, 2, ..., n}. The bipartite Kneser graph H(n, k) is the graph with the all k-element and all (n − k)-element subsets of [n] as vertices, and there is an edge between any two vertices, when one is a subset of the other. In this paper, we show that H(n, k) is an arc-transitive graph. Also, we show that H(n, 1) is a distance-transitive Cayley graph. Finally, we determine the automorphism group of the graph H(n, 1) and show that Aut(H(n, 1)) ∼ = Sym([n]) × Z 2 , where Z 2 is the cyclic group of order 2. Moreover, we pose some open problems about the automorphism group of the bipartite Kneser graph H(n, k).

The line graph of the crown graph is distance integral

“…Remark 2.7. Although the crown graph Cr(n) is a distance-transitive graph (and consequently it is distance-regular [4,9]), it is easy to check that the graph L(Cr(n)) is not distance-regular. Hence we can not use the theory of distance -regular graphs for determining the set of distance eigenvalues of this graph.…”

“…Let β be the function on the vertex-set of the graph L(Cr(n)) defined by the rule, β(i, j) = (j, i). Now we can check that Aut(L(Cr(n))) ∼ = Sym([n]) × β [9,11,13], where β is the subgroup generated by the automorphism β in the automorphism group of the graph L(Cr(n)).…”

“…. , n}, in which two vertices v and w are adjacent if and only if v ⊂ w or w ⊂ v. Recently, this class of graphs has been studied form several aspects [9,11,12,13]. It is easy to see that the crown graph Cr(n) is isomorphic with the bipartite graph H(n, 1).…”

On the distance eigenvalues of design graphs

“…The bipartite Kneser graph H(n, 1) has V as its vertex-set, and two vertices v, w are adjacent if and only if v ⊂ w or w ⊂ v. It is easy to see that H(n, 1) is a bipartite graph of diameter 3. Some of properties of the graph H(n, 1) and a generalization of it have been appeared in [11,13,14]. We can easyly show that H(n, 1) is a design graph with parameters (n, n − 1, n − 2).…”

The automorphism group of the bipartite Kneser graph

“…The regular hyperstar graph Q 2n+1 (n, n + 1) has been investigated from various aspects, by various authors and some of the recent works about this class of graphs are [3,6,11,13,16,17]. The following figure shows the graph H(5, 2) (Q 5 (2,3)) in plane.…”