LAUSR

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