There are many bounds for the spectral radius of graphs in terms of no. of vertices, maximum degree, chromatic number etc. I wish to know till date what are the best lower and upper bound for the spectral radius of a graph? 



best known bounds for spectral radius
Actually you asked right question, but it is so vague to answer. Fortunately, there is a good book which you can find it very interesting. The book is:


"Spectral Radius of Graphs" by Dragan Stevanovic.[Find the book here](http://store.elsevier.com/Spectral-Radius-of-Graphs/Dragan-Stevanovic/isbn-9780128020685/)


There are a lot of best possible bound for the graphs based on their structures and special parameters. Also, there are a lot of conjectures which shows we do not know the best possible bound in many cases. For example, we know that if $G$ is planar with $n\geq 3$ vertices, then we have $\lambda\_1\leq \sqrt{3n-9}+4$. But it conjectured that for planar graph with $n$ vertices we have:
$$\lambda\_1(G)\leq \lambda\_1(K\_2\vee P\_{n-2}).$$


If you study this book, you will find that why this question is so general and also so difficult. 


