[論文レビュー] A survey on spectral conditions for some extremal graph problems

Yongtao Li, Weijun Liu+1 more|arXiv (Cornell University)|Nov 5, 2021

Graph theory and applications参考文献 155被引用数 29

ひとこと要約

A comprehensive survey of spectral extremal results for Turán-type problems and related graph properties, focusing on adjacency and signless Laplacian spectra.

ABSTRACT

This survey is two-fold. We first report new progress on the spectral extremal results on the Turán type problems in graph theory. More precisely, we shall summarize the spectral Turán function in terms of the adjacency spectral radius and the signless Laplacian spectral radius for various graphs. For instance, the complete graphs, general graphs with chromatic number at least three, complete bipartite graphs, odd cycles, even cycles, color-critical graphs and intersecting triangles. The second goal is to conclude some recent results of the spectral conditions on some graphical properties. By a unified method, we present some sufficient conditions based on the adjacency spectral radius and the signless Laplacian spectral radius for a graph to be Hamiltonian, $k$-Hamiltonian, $k$-edge-Hamiltonian, traceable, $k$-path-coverable, $k$-connected, $k$-edge-connected, Hamilton-connected, perfect matching and $β$-deficient.

研究の動機と目的

• Summarize spectral (adjacency and signless Laplacian) Turán-type results for complete graphs, general graphs, bipartite graphs, cycles, and color-critical graphs.
• Survey how spectral parameters constrain graphical properties like Hamiltonicity, connectivity, and perfect matchings.
• Highlight unified methods linking spectral bounds to classical extremal results and identify key extremal structures (e.g., Turán graphs).

提案手法

• Present spectral analogues of Turán-type theorems using eigenvalues and the Motzkin–Straus Lagrangian framework.
• Discuss bounds such as λ(G) ≤ λ(T_r(n)) for K_{r+1}-free graphs and q(G) ≤ q(T_r(n)) for signless Laplacian spectra.
• Leverage classical results (Nosal, Wilf, Erdős–Stone–Simonovits) to relate spectral bounds to edge counts.
• Outline p-spectral radius generalizations and methods avoiding characteristic polynomials.
• Reference and synthesize results proving when equality occurs and which graphs attain extremality.]
• research_questions: [

実験結果

リサーチクエスチョン

• RQ1 What are the maximal adjacency spectral radius and signless Laplacian spectral radius for graphs not containing a given subgraph F?
• RQ2 How do spectral extremal results compare to classical edge-extremal (Turán-type) results across various forbidden subgraphs?
• RQ3 Which graph classes (e.g., K_{r+1}-free, bipartite, cycles) attain equality in spectral Turán-type bounds?
• RQ4 How do spectral bounds imply or strengthen traditional Turán-type edge bounds?
• RQ5 What unified methods connect spectral radii to Hamiltonicity, connectivity, and matching properties?

主な発見

• Spectral Turán results show λ(G) ≤ λ(T_r(n)) for K_{r+1}-free graphs, with equality characterizing T_r(n) in certain cases.
• Signless Laplacian bounds q(G) ≤ q(T_r(n)) extend Wilf-type results to K_{r+1}-free graphs, with equality in specific configurations.
• Motzkin–Straus framework links clique number to spectral bounds, enabling spectral proofs of Turán-type results.
• Bounds such as λ(G) ≤ (1 − 1/χ(F))n and related refinements connect spectral radius to chromatic number and forbidden subgraphs.
• Extensions to p-spectral radius provide a broader, polynomial-based perspective on spectral extremality.