Abstract
In this paper, we study the symmetry of the nonzero component graph \(\Gamma _n ({\mathbb {F}})\) of an n-dimensional vector space over a field \({\mathbb {F}}\). We explicitly compute the automorphism group of \(\Gamma _n ({\mathbb {F}})\) and compute stabilizers, orbits, and the determining number of \(\Gamma _n ({\mathbb {F}})\). We then characterize all the determining sets of \(\Gamma _n ({\mathbb {F}})\). Using this characterization, we compute the determining polynomial of \(\Gamma _n ({\mathbb {F}})\) when \({\mathbb {F}}\) is a finite field. We also discuss the core property and the Wiener index of \(\Gamma _n ({\mathbb {F}})\).
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Communicated by Wen Chean Teh.
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The first author expresses gratitude to Prof. S. Sundar from The Institute of Mathematical Sciences, Chennai, for the valuable insights into mathematics and to Dr. Selvaraja for the research grant at the Chennai Mathematical Institute during part of this work. We are very thankful to the anonymous referees for their various comments and suggestions to improve the article.
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Murugan, S.P., Manikandan, S. & Selvakumar, A. The Automorphism Group of Nonzero Component Graph of Vector Space. Bull. Malays. Math. Sci. Soc. 47, 56 (2024). https://doi.org/10.1007/s40840-024-01651-1
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DOI: https://doi.org/10.1007/s40840-024-01651-1