It is well-known that the largest eigenvalue $\lambda_{\max}$ of the adjacency matrix of a graph $G$ lies between the average and the maximal degree of $G$. Another known lower estimate is due to Mohar and says that 
$$\lambda_{\max}\ge\sqrt{\deg_{\max}}.$$ 
The reason for this is that $\lambda_{\max}$ is motononic wrt graph inclusion, so one could pick a vertex of maximal degree and all the incident edges and delete the rest of the graph; since this star has maximal adjacency eigenvalue $\sqrt{\deg_{\max}}$, we are done.

What about line graphs? My feeling is that this estimate cannot be sharp, since line graphs are claw-free. What is the "minimal induced subgraph" around a vertex of maximal degree in a line graph? Can the above estimate be improved?