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One can almost never hear the shape of a digraph

Da Zhao

Abstract

Kac raised the question `Can one hear the shape of a drum?' The answer is negative. On the other hand, one can hear the shape of a drum in the generic case. Haemers conjectured that almost all graphs are determined by their spectra, which can be regarded as a discrete version of Kac's question. Vu conjectured a similar phenomenon for symmetric $\pm 1$ matrices. The first main result of this paper shows that one can almost never hear the shape of a digraph by proving that almost all digraphs are not isomorphic to their reverse and almost all digraphs have trivial automorphism groups. This indicates that the real symmetric condition is vital in both Haemers' conjecture and Vu's conjecture. The second main result of this paper shows that almost all graphs of order $n$ have no cospectral mates with height $o(( n / \ln n)^{1/10})$, improving an earlier result on cospectral mates with fixed level.

One can almost never hear the shape of a digraph

Abstract

Kac raised the question `Can one hear the shape of a drum?' The answer is negative. On the other hand, one can hear the shape of a drum in the generic case. Haemers conjectured that almost all graphs are determined by their spectra, which can be regarded as a discrete version of Kac's question. Vu conjectured a similar phenomenon for symmetric matrices. The first main result of this paper shows that one can almost never hear the shape of a digraph by proving that almost all digraphs are not isomorphic to their reverse and almost all digraphs have trivial automorphism groups. This indicates that the real symmetric condition is vital in both Haemers' conjecture and Vu's conjecture. The second main result of this paper shows that almost all graphs of order have no cospectral mates with height , improving an earlier result on cospectral mates with fixed level.

Paper Structure

This paper contains 4 sections, 8 theorems, 34 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of \ref{['thm:non-isomorphic-to-reverse']}
  3. Proof of \ref{['thm:fixed-height']}
  4. Discussion

Key Result

Theorem 1.1

One can almost never hear the shape of a digraph. Namely, let $n$ be a positive integer, $0 < p < 1$, and $D \sim \mathcal{D}(n, p)$ a random digraph, then equivalently as $n \to \infty$. Here the spectrum can be taken as adjacency spectrum or Hermitian adjacency spectrum. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • proof : Proof of \ref{['thm:non-isomorphic-to-reverse']}
  • Lemma 3.1
  • proof
  • Claim 3.2
  • ...and 9 more