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4 fixed a typo

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 $\lambda_n-\lambda_{n+1})$ \lambda_n-\lambda_{n+1}$in Hueckel theory is called the HOMO-LUMO gap. Edit: Note that in the original formula for$R(G)$the absolute value signs were missing. 3 corrected a misprint in formula Let$ \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$R(G) = max(\lambda_n,\lambda_{n+1})$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.

2 corrected spelling

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 Hueckle thoery Hueckel theory is called the HOMO-LUMO gap.

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