On the automorphism groups of connected bipartite irreducible graphs

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 $G\mathrm{.}$ The graph $G$ is called $irreducible$ whenever for every $v,w\in V$ if $v\mathrm{\ne }w$, then $N(v)\mathrm{\ne }N(w)\mathrm{.}$ 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$v,w$areadjacent,whereas$c(v,w) \neq a_0$,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 $Aut(G) \times $\mathbb{Z}$_2$\mathrm{.}Finally,weshowthattheJohnsongraph$J(n,k)$ is a stable graph.