Spectral Extremal Results with Forbidding Linear Forests

The Turan type extremal problems ask to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, their spectral counterparts ask to maximize spectral radius of all graphs which do not contain fixed subgraphs. In this paper, we determine the maximum spectral radius of all graphs without a linear forest as a subgraph and all the extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without kP3 as a subgraph are obtained and all the extremal graphs are also determined. Moreover, some relations between Turan type extremal problems and their spectral counterparts are discussed.