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Unsolved Problems in Spectral Graph Theory

Lele Liu, Bo Ning

TL;DR

This paper presents a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures, primarily on the adjacency matrix of graphs.

Abstract

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.

Unsolved Problems in Spectral Graph Theory

TL;DR

This paper presents a collection of topics in spectral graph theory, covering a range of open problems and conjectures, primarily on the adjacency matrix of graphs.

Abstract

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.
Paper Structure (24 sections, 5 theorems, 34 equations, 1 figure)

This paper contains 24 sections, 5 theorems, 34 equations, 1 figure.

Table of Contents

  1. Extensions of two classic inequalities
  2. An extension of Hong's inequality
  3. An extension of Wilf's inequality
  4. The Bollobás-Nikiforov Conjecture
  5. Maximum spectral radius of planar (hyper)graphs
  6. The relationship between Turán theorem and spectral Turán theorem
  7. Tight spectral conditions for a cycle of given length
  8. Nikiforov's problem on consecutive cycles
  9. Graph toughness from Laplacian eigenvalues
  10. Hamilton cycles in pseudo-random graphs
  11. A spectral problem on counting subgraphs
  12. Extreme eigenvalues of nonregular graphs
  13. Graphs with a prescribed average degree
  14. The Bilu-Linial Conjecture
  15. Isoperimetric problem in hypercube
  16. ...and 9 more sections

Key Result

Theorem 1

Let $t\in\mathbb{N}$ and $r\geq -\binom{t+1}{2}$. If $G$ is a graph on $n$ vertices with property $P_{t,r}$, then and asymptotically, Furthermore, the asymptotic bound is tight.

Figures (1)

  • Figure 1: The double kite graph $DK(8,5)$

Theorems & Definitions (30)

  • Conjecture 1: Elphick-Farber-Goldberg-Wocjan EFGW16
  • Conjecture 2: Elphick-Wocjan Elphick-Wocjan2018
  • Conjecture 3: Bollobás-Nikiforov BN07
  • Conjecture 4: Elphick-Linz-Wocjan Elphick-Linz-Wocjan2021
  • Conjecture 5: Boots-Royle 1991 Boots-Royle1991 and independently Cao-Vince 1993 Cao-Vince1993
  • Conjecture 6: Ellingham-Lu-Wang ELW22
  • Theorem 1: G96
  • Conjecture 7: Zhai-Lin-Shu Zhai-Lin-Shu2021
  • Conjecture 8: Haemers Haemers2020
  • Conjecture 9: Krivelevich-Sudakov Krivelevich-Sudakov2003
  • ...and 20 more