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A new conjecture on the inertia of graphs

Saieed Akbari, Clive Elphick, Hitesh Kumar, Shivaramakrishna Pragada, Quanyu Tang

TL;DR

The paper introduces a universal inertia bound $2n^+(G) \le n^-(G)(n^-(G)+1)$ for graphs, motivated by the strong bounds for strongly regular graphs and connected to Torgašev's problem. It develops a framework of reductions and spectral techniques, then verifies the conjecture across broad classes, including random graphs, graphs with cut vertices, subquartic and planar graphs, line graphs, self-complementary graphs, graphs with few odd cycles, graph products, and cographs. Key methods include interlacing, neighborhood-deletion, and relations between line graphs and Laplacian spectra, yielding both general results and numerous equality cases. The work highlights extensive supporting evidence, discusses infinite families of tight examples via twins, and raises open questions about potential polynomial bounds and stronger line-graph statements.

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torgašev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the conjecture is exact. We also conjecture that for any connected graph $G$, its line graph $L(G)$ satisfies $n^+(L(G)) \le n^-(L(G)) + 1$, and obtain partial results.

A new conjecture on the inertia of graphs

TL;DR

The paper introduces a universal inertia bound for graphs, motivated by the strong bounds for strongly regular graphs and connected to Torgašev's problem. It develops a framework of reductions and spectral techniques, then verifies the conjecture across broad classes, including random graphs, graphs with cut vertices, subquartic and planar graphs, line graphs, self-complementary graphs, graphs with few odd cycles, graph products, and cographs. Key methods include interlacing, neighborhood-deletion, and relations between line graphs and Laplacian spectra, yielding both general results and numerous equality cases. The work highlights extensive supporting evidence, discusses infinite families of tight examples via twins, and raises open questions about potential polynomial bounds and stronger line-graph statements.

Abstract

Let be a graph with adjacency matrix . We conjecture that where and denote the number of positive and negative eigenvalues of , respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torgašev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the conjecture is exact. We also conjecture that for any connected graph , its line graph satisfies , and obtain partial results.

Paper Structure

This paper contains 13 sections, 26 theorems, 57 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Torgašev's problem
  3. A coarse upper bound
  4. Special graph families
  5. Random graphs
  6. Graphs with a cut vertex
  7. Subquartic graphs and planar graphs
  8. Line graphs
  9. Self-complementary graphs
  10. Graphs with few odd cycles
  11. Graph products
  12. Cographs
  13. Concluding remarks

Key Result

Theorem 1.1

Let $G$ be a regular graph of order $n$ with least eigenvalue $\lambda$. If $\lambda < -1$ and multiplicity of $\lambda$ is $n-k$ for some $k\ge 1$, then $n\le \frac{k(k+1)}{2}-1$.

Figures (2)

  • Figure 1: The graphs $H_1$ and $H_2$, both satisfying $n^- = 3$ and $n^+ = 6$.
  • Figure 2: 5-prism

Theorems & Definitions (47)

  • Theorem 1.1: Seidel_1977 cf. Seidel_1979Brouwer_Haemers_book
  • Theorem 1.2: Bell_Rowlinson_2003
  • Theorem 1.3: Absolute bound for SRGs, cf. Seidel_1979
  • Conjecture 1.4
  • Conjecture 2.1: Akbari_Cameron_Khosrovshahi
  • Theorem 2.2: Torgašev's Theorem Torgasev_1985
  • Remark
  • Conjecture 2.3: Mohammadian_2022
  • Theorem 2.4: Kotlov_Lovasz_1996
  • Conjecture 2.5: Ma_Yang_Li_2013
  • ...and 37 more