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Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?

By "effective" I mean something that would let one iterate through all such graphs with reasonably few (say, a few dozens) vertices, or, better yet, to prove theorems about such graphs :) (Ideally, a list similar to that of Dynkin diagrams.)

The question is motivated by this one: "spectrum of an adjacency matrixspectrum of an adjacency matrix" and by my own research. It seems wild, but I'm not an expert.

Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?

By "effective" I mean something that would let one iterate through all such graphs with reasonably few (say, a few dozens) vertices, or, better yet, to prove theorems about such graphs :) (Ideally, a list similar to that of Dynkin diagrams.)

The question is motivated by this one: "spectrum of an adjacency matrix" and by my own research. It seems wild, but I'm not an expert.

Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?

By "effective" I mean something that would let one iterate through all such graphs with reasonably few (say, a few dozens) vertices, or, better yet, to prove theorems about such graphs :) (Ideally, a list similar to that of Dynkin diagrams.)

The question is motivated by this one: "spectrum of an adjacency matrix" and by my own research. It seems wild, but I'm not an expert.

edited title (name used in the literature)
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Alex Degtyarev
Alex Degtyarev
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Hyperbolic Reflexive (hyperbolic) graphs

The question is motivated by this one: "spectrum of an adjacency matrix" and by my own research. It seems wild, but I'm not an expert.

Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?

By "effective" I mean something that would let one iterate through all such graphs with reasonably few (say, a few dozens) vertices, or, better yet, to prove theorems about such graphs :) (Ideally, a list similar to that of Dynkin diagrams.)

The question is motivated by this one: "spectrum of an adjacency matrix" and by my own research. It seems wild, but I'm not an expert.

Hyperbolic graphs

The question is motivated by this one: "spectrum of an adjacency matrix" and by my own research. It seems wild, but I'm not an expert.

Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?

By "effective" I mean something that would let one iterate through all such graphs with reasonably few (say, a few dozens) vertices, or, better yet, to prove theorems about such graphs :) (Ideally, a list similar to that of Dynkin diagrams.)

Reflexive (hyperbolic) graphs

Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?

By "effective" I mean something that would let one iterate through all such graphs with reasonably few (say, a few dozens) vertices, or, better yet, to prove theorems about such graphs :) (Ideally, a list similar to that of Dynkin diagrams.)

The question is motivated by this one: "spectrum of an adjacency matrix" and by my own research. It seems wild, but I'm not an expert.

Source Link
Alex Degtyarev
Alex Degtyarev
  • 5k
  • 5
  • 23
  • 26

Hyperbolic graphs

The question is motivated by this one: "spectrum of an adjacency matrix" and by my own research. It seems wild, but I'm not an expert.

Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?

By "effective" I mean something that would let one iterate through all such graphs with reasonably few (say, a few dozens) vertices, or, better yet, to prove theorems about such graphs :) (Ideally, a list similar to that of Dynkin diagrams.)