Automorphism group of a family of distance-regular graphs which are not distance-transitive

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Lorestan University, Khorramabad, Iran

2 Department of Mathematics, Presidency University, Kolkata, India

Abstract

Let $G_n=\mathbb{Z}_n\times \mathbb{Z}_n$ for $n\geq 4$ and $S=\{(i,0),(0,i),(i,i): 1\leq i \leq n-1\}\subset G_n$. Define $\Gamma(n)$ to be the Cayley graph of $G_n$ with respect to the connecting set $S$. It is known that $\Gamma(n)$ is a strongly regular graph with the parameters $(n^2, 3n-3, n, 6)$ \cite{19}. Hence $\Gamma(n)$ is a distance-regular graph. It is known that every distance-transitive graph is distance-regular, but the converse is not true. In this paper, we study some algebraic properties of the graph $\Gamma(n)$. Then by determining the automorphism group of this family of graphs, we show that the graphs under study are not distance-transitive.

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References

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Articles in Press, Corrected Proof
Available Online from 10 January 2025

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History

  • Receive Date: 06 August 2024
  • Revise Date: 03 January 2025
  • Accept Date: 03 January 2025
  • Published Online: 10 January 2025

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