Let S+(G) denote the sum of the squares of the positive eigenvalues of the adjacency matrix of a graph G. Let m denote the number of edges and q the chromatic number. Is it true that S+(G)is less than or equal to 2m(q-1)/q?

This conjecture is exact for all bipartite graphs and correct for strongly regular, complete q-partite, regular two-graphs and (it appears) all Kneser graphs.  

The chromatic number cannot be replaced with the clique number in this conjecture, because the Coxeter graph provides a counter-example. Experimentally this bound is sometimes better and sometimes worse than the Hoffman lower bound for the chromatic number.

It is not hard to prove, using a result due to Cvetkovic, that S+(G) is less than or equal to 2m(n - a)/(n - a +1), where n is the number of vertices and a the independence number of G. It is well known that (q -1) is less than or equal to (n - a). It is also known that the square of the largest eigenvalue is less than or equal to 2m(q-1)/q.

Let S-(G) denote the sum of the squares of the negative eigenvalues of G. Recalling that 
S+(G) + S-(G) = 2m, the above conjecture can be re-arranged as 1 + S+(G)/S-(G) is less than or equal to q. This has the same form as the Hoffman lower bound for q.