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Let $ \lambda_1 \ge \lambda_2 \ge \dots \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 $R(G) = max(|\lambda_n|,|\lambda_{n+1}|)$ 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.

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.

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

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.

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

Let $ \lambda_1 \ge \lambda_2 \ge \dots \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 $R(G) = max(|\lambda_n|,|\lambda_{n+1}|)$ 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.

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.

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

Let $ \lambda_1 \ge \lambda_2 \ge \dots \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 $R(G) = max(|\lambda_n|,|\lambda_{n+1}|)$ 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.

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.

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

Let $ \lambda_1 \ge \lambda_2 \ge \dots \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 $R(G) = max(\lambda_n,\lambda_{n+1})$ 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.

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.

Let $ \lambda_1 \ge \lambda_2 \ge \dots \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 $R(G) = max(|\lambda_n|,|\lambda_{n+1}|)$ 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.

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.

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