On the automorphism groups of connected bipartite irreducible graphs

Abstract

Let $$G=(V,E)$$ be a graph with the vertex-set V and the edge-set E. Let N(v) denote the set of neighbors of the vertex v of G. The graph G is called irreducible whenever for every $$v,w \in V$$ if $$v \ne w$$ , then $$N(v)\ne N(w).$$ In this paper, we present a method for finding automorphism groups of connected bipartite irreducible graphs. Then, by our method, we determine automorphism groups of some classes of connected bipartite irreducible graphs, including a class of graphs which are derived from Grassmann graphs. Let $$a_0$$ be a fixed positive integer. We show that if G is a connected non-bipartite irreducible graph such that $$c(v,w)=|N(v)\cap N(w)|=a_0$$ when v, w are adjacent, whereas $$c(v,w) \ne a_0$$ , when v, w are not adjacent, then G is a stable graph, that is, the automorphism group of the bipartite double cover of G is isomorphic with the group $${\mathrm{Aut}}(G) \times {\mathbb {Z}}_2$$ . Finally, we show that the Johnson graph J(n, k) is a stable graph.