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The least balanced graphs and trees

Péter Csikvári, Viktor Harangi

TL;DR

This work resolves a longstanding extremal spectral question: among n-vertex connected graphs, the smallest value of Γ_G = (||x||_1)^2/(||x||_2)^2, with x the Perron vector, is achieved by G = K_4 + P_{n-4} for n ≥ 6, and among n-vertex trees by T = S_5 + P_{n-5} for n ≥ 8. The authors develop a robust framework based on heavy subgraphs and a resolvent-extended kernel method to bound Γ_G from below by analyzing a small kernel around the master vertex and a single-path tail, aided by computer-assisted verification to exhaust all kernels. They derive explicit limiting ratios β_* = Γ_{K_4+P_∞} = (5+3√3)/2 and β_tr = Γ_{S_5+P_∞} = 4+2√3, and prove that any deviation from the conjectured kernels forces Γ_G to exceed these thresholds. The results establish precise extremal structures and demonstrate a powerful toolkit for spectral-extremal problems on graphs and trees, with potential applicability to related Perron-vector concentration questions.

Abstract

Given a connected graph, the principal eigenvector of the adjacency matrix (often called the Perron vector) can be used to assign positive weights to the vertices. A natural way to measure the homogeneousness of this vector is by considering the ratio of its $\ell^1$ and $\ell^2$ norms. It is easy to see that the most balanced graphs in this sense (i.e., the ones with the largest ratio) are the regular graphs. What about the least balanced graphs with the smallest ratio? It was conjectured by Rücker, Rücker and Gutman that, for any given $n \geq 6$, among $n$-vertex connected graphs the smallest ratio is achieved by the complete graph $K_4$ with a single path $P_{n-4}$ attached to one of its vertices. In this paper we confirm this conjecture. We also verify the analogous conjecture for trees: for any given $n \geq 8$, among $n$-vertex trees the smallest ratio is achieved by the star graph $S_5$ with a path $P_{n-5}$ attached to its central vertex.

The least balanced graphs and trees

TL;DR

This work resolves a longstanding extremal spectral question: among n-vertex connected graphs, the smallest value of Γ_G = (||x||_1)^2/(||x||_2)^2, with x the Perron vector, is achieved by G = K_4 + P_{n-4} for n ≥ 6, and among n-vertex trees by T = S_5 + P_{n-5} for n ≥ 8. The authors develop a robust framework based on heavy subgraphs and a resolvent-extended kernel method to bound Γ_G from below by analyzing a small kernel around the master vertex and a single-path tail, aided by computer-assisted verification to exhaust all kernels. They derive explicit limiting ratios β_* = Γ_{K_4+P_∞} = (5+3√3)/2 and β_tr = Γ_{S_5+P_∞} = 4+2√3, and prove that any deviation from the conjectured kernels forces Γ_G to exceed these thresholds. The results establish precise extremal structures and demonstrate a powerful toolkit for spectral-extremal problems on graphs and trees, with potential applicability to related Perron-vector concentration questions.

Abstract

Given a connected graph, the principal eigenvector of the adjacency matrix (often called the Perron vector) can be used to assign positive weights to the vertices. A natural way to measure the homogeneousness of this vector is by considering the ratio of its and norms. It is easy to see that the most balanced graphs in this sense (i.e., the ones with the largest ratio) are the regular graphs. What about the least balanced graphs with the smallest ratio? It was conjectured by Rücker, Rücker and Gutman that, for any given , among -vertex connected graphs the smallest ratio is achieved by the complete graph with a single path attached to one of its vertices. In this paper we confirm this conjecture. We also verify the analogous conjecture for trees: for any given , among -vertex trees the smallest ratio is achieved by the star graph with a path attached to its central vertex.

Paper Structure

This paper contains 29 sections, 33 theorems, 216 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our approach
  3. Notations.
  4. Our key bounding method
  5. Outline of the proof of Theorem \ref{['thm:only_K4Pk']}
  6. Program code
  7. Basic results and estimates
  8. Resolvent matrix
  9. Graphs with largest eigenvalue at most 2
  10. Heavy subgraphs and the degree of the master vertex
  11. Degree of the master vertex.
  12. Towards better heaviness constants
  13. Bounding via heavy subgraphs
  14. Determining the optimal kernel with a computer
  15. Connected graphs
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $n\geq 7$ and $G$ be an $n$-vertex connected graph. Then In particular, among $n$-vertex connected graphs, $\Gamma_G$ is minimized by $G = K_4 + P_{n-4}$. (This latter statement holds for $n\geq 6$ already.)

Figures (2)

  • Figure 1: If $\Gamma_G<5.25$, then $G$ contains a $K_4$ that is joined to the remaining part $G'$ by a single edge incident to the master vertex $o$.
  • Figure 2: Comparison of our bounds as functions of $\lambda$

Theorems & Definitions (81)

  • Conjecture 1.1: Rücker, Rücker and Gutman rucker2002kites
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • proof : Proof outline
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 71 more