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https://www.ias.ac.in/article/fulltext/pmsc/130/0057

Automorphism group; bipartite double cover of a graph; Grassmann graph; stable graph; Johnson graph.

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

S MORTEZA MIRAFZAL¹

1. Department of Mathematics, Lorestan University, Khorramabad, Iran

Published

19 September 2020

Final version published online

19 September 2020