A survey on spectral conditions for some extremal graph problems

TL;DR

This survey reviews recent advances in spectral graph theory, focusing on extremal problems related to Turán-type bounds and spectral conditions for various graph properties, using adjacency and signless Laplacian spectra.

Contribution

It summarizes new spectral extremal results for Turán problems and provides unified spectral conditions for multiple graph properties, enhancing understanding of spectral graph theory.

Findings

Spectral Turán functions characterized for various graphs.

Sufficient spectral conditions for Hamiltonian and related properties.

Unified approach to spectral conditions across multiple graph properties.

Abstract

This survey is two-fold. We first report new progress on the spectral extremal results on the Tur\'{a}n type problems in graph theory. More precisely, we shall summarize the spectral Tur\'{a}n 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 $\beta$-deficient.