Let $ \lambda\_1 \ge \lambda\_2 \ge \dots \ge \lambda\_{2n} $
be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references that would consider the middle eigenvalues $\lambda\_n$ and $\lambda\_{n+1}$. In particular, the bounds on $R(G) = \max \left( |\lambda\_n|, |\lambda\_{n+1}| \right)$ are welcome.


For instance, a computer search showed that most connected graphs with maximum valence $3$ have $R(G) \le 1$. The only known exception is the Heawood graph.


The motivation for this question comes from theoretical chemistry, where the difference $\lambda\_n - \lambda\_{n+1}$ in Hueckel theory is called the [HOMO-LUMO gap](https://en.wikipedia.org/wiki/HOMO_and_LUMO).


**Edit:** Note that in the original formula for $R(G)$ the absolute value signs were missing.



The middle eigenvalues of an undirected graph
One thing that you might find useful is the [Cauchy interlacing theorem](https://en.wikipedia.org/wiki/Min-max_theorem#Cauchy_interlacing_theorem).


In response to the comment, presumably Tomaz's interest is in some particular sorts of graphs. It may be the case that such a graph has some well-understood subgraphs. Then you could reduce the question about the size of the middle eigenvalues of the big graph to a question about the size of the "approximately middle" eigenvalues of the smaller graph.


