Distinguishing Number of Non-Zero Component Graphs

Abstract: A non-zero component graph $G(\mathbb{V})$ associated to a finite vector space $\mathbb{V}$ is a graph whose vertices are non-zero vectors of $\mathbb{V}$ and two vertices are adjacent, if their corresponding vectors have at least one non-zero component common in their linear combination of basis vectors. In this paper, we extend the study of properties of automorphisms of non-zero component graphs. We prove that every permutation of basis vectors can be extended to an automorphism of $G(\mathbb{V})$. We prove that the symmetric group of basis vectors of $\mathbb{V}$ is isomorphic to the automorphism group of $G(\mathbb{V})$. We find the distinguishing number of the graph for both of the cases, when the number of field elements of vector space $\mathbb{V}$ are 2 or more than 2.