# Bounds on spectral radius using chromatic number

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 $$n,m$$

• There are bipartite graphs with arbitrarily high spectral radius, so you cannot use chromatic number on its own. – Gordon Royle May 5 at 11:08
• @GordonRoyle;cant we give a lower bound using chromatic numbers too? – Learnmore May 5 at 11:11
• Possibly $\rho(A)\geq\chi-1$? – Bullet51 May 5 at 15:42