The Maximum Spectral Radius of Graphs without Spanning Linear Forests

Given a family F of graphs, a graph G is called F-free if G contains none of F as its subgraph. The following problem is one of the most concerned problems in spectral extremal graph theory: what is the maximum spectral radius of an n-vertex F-free graph? If each connected component of a graph is either a path (star) or an isolated vertex, then we call it a linear (star) forest. Denote by L_n,k and S_n,k the family of all n-vertex linear forests and star forests with k edges, respectively. In this paper, we obtain the maximum spectral radius of an n-vertex L_n,k-free graph and characterize the extremal graphs based on Kelmans transformation. Also, we obtain the maximum spectral radius of an n-vertex S_n,k-free graph and characterize the unique extremal graph.