Domination Numbers and Automorphisms of Dual Graphs Over Vector Spaces

Abstract

Let $F_q$ be a finite field of q elements, $\mathbb {V}$ an n-dimensional vector space over $F_q$, and $\mathbb {V}^*$ the dual space of $\mathbb {V}$, i.e., the vector space of all linear function over $\mathbb {V}$. The graph $\hbox {DG}(\mathbb {V})$, called the dual graph of $\mathbb {V}$, is defined to be a bipartite graph, whose vertex set is partitioned into two coloring sets, respectively, consisting of all one-dimensional subspaces of $\mathbb {V}$ and all one-dimensional subspaces of $\mathbb {V}^*$, and there is an undirected edge between an one-dimensional subspace [v] of $\mathbb {V}$ and an one-dimensional subspace [f] of $\mathbb {V}^*$ if and only if $f(v) = 0$. In this paper, the domination number, independence number, diameter and girth of $\hbox {DG}(\mathbb {V})$ are, respectively, determined; some automorphisms of $\hbox {DG}(\mathbb {V})$ are introduced, and such a graph is proved to be distance transitive.