In this thesis, we study relationships between linear star number and star number and obtain bounds on the linear star number. We obtain an upper bound on linear star number in term of star number：s*(G) ≦ 3s(G). When we forbid certain induced subgraphs, we obtain an upper bound on linear star number. If G is a graph without induced K4－e., we prove that s*(G) ≦ s(G)+1. And, the linear star number of the triangle-free graph is also bounded by s(G)+1. The linear star number and star number are equal when G is a graph with △(G)=3. When G is a graph with △(G)=4, we also obtain s*(G)≦s(G)+1.