The Automorphism Group of Nonzero Component Graph of Vector Space

In this paper, we study the symmetry of the nonzero component graph \(\Gamma _n ({\mathbb {F}})\) ${\mathrm{\Gamma }}_n(\mathbb{F})$ of an n-dimensional vector space over a field \({\mathbb {F}}\) $\mathbb{F}$ . We explicitly compute the automorphism group of \(\Gamma _n ({\mathbb {F}})\) ${\mathrm{\Gamma }}_n(\mathbb{F})$ and compute stabilizers, orbits, and the determining number of \(\Gamma _n ({\mathbb {F}})\) ${\mathrm{\Gamma }}_n(\mathbb{F})$ . We then characterize all the determining sets of \(\Gamma _n ({\mathbb {F}})\) ${\mathrm{\Gamma }}_n(\mathbb{F})$ . Using this characterization, we compute the determining polynomial of \(\Gamma _n ({\mathbb {F}})\) ${\mathrm{\Gamma }}_n(\mathbb{F})$ when \({\mathbb {F}}\) $\mathbb{F}$ is a finite field. We also discuss the core property and the Wiener index of \(\Gamma _n ({\mathbb {F}})\) ${\mathrm{\Gamma }}_n(\mathbb{F})$ .