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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?

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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

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.

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