I am struggling with this question:

If I have a connected graph $G$ on $n$ vertices and $m$ edges with chromatic number $d$ then how can I give a bound(lower and upper) on its spectral radius in terms of its chromatic number.

There are some properties on the largest eigen value of adjacency matrix $A$ of a graph $G$.

If a Graph $G$ is connected then the largest eigen value of $A$ say $\rho(A)$ is positive and it has a positive eigen vector(Perron-Frobenius Theorem)

How can I give an upper bound on $\rho(A)$ in terms of chromatic number of $G$?

I want the bound to be proved in terms of chromatic number and in terms of number of vertices and edges of the graph $G$.

How can I do it?

**Note:I checked Wilf's and Hoffmans bound.But I want to find some other bounds which involve using the Perrons Theorem and uses $n,m$**