On the automorphism groups of connected bipartite irreducible graphs

TL;DR

This paper introduces a method to determine automorphism groups of connected bipartite irreducible graphs, applies it to specific classes including Grassmann-derived graphs, and explores properties of stable graphs and Johnson graphs.

Contribution

The paper presents a novel method for finding automorphism groups of connected bipartite irreducible graphs and characterizes stability in certain classes, including Johnson graphs.

Findings

Automorphism groups of certain bipartite irreducible graphs are determined.

Connected bipartite irreducible graphs derived from Grassmann graphs are analyzed.

Johnson graph J(n,k) is shown to be a stable graph.

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 $v \neq w$, then $N(v)\neq N(w).$ 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 $c(v,w) \neq a_0$, 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 $Aut(G) \times \mathbb{Z}_2$. Finally, we show that the Johnson graph $J(n,k)$ is a stable graph.