Spectral extremal problems for degenerate graphs

TL;DR

This paper explores spectral extremal problems for degenerate graph families, establishing stability results and characterizations that unify and extend existing theorems in spectral graph theory.

Contribution

It introduces new spectral stability results and characterizations for extremal graphs within degenerate graph families, broadening the understanding of spectral extremal problems.

Findings

Established spectral stability for degenerate graph families.

Characterized spectral extremal graphs for broad classes of graphs.

Connected extremal and spectral extremal graphs for these families.

Abstract

A family of graphs is called degenerate if it contains at least one bipartite graph. In this paper, we investigate the spectral extremal problems for a degenerate family of graphs $\mathcal{F}$. By employing covering and independent covering of graphs, we establish a spectral stability result for $\mathcal{F}$. Using this stability result, we prove two general theorems that characterize spectral extremal graphs for a broad class of graph families $\mathcal{F}$ and imply several new and known results. Meanwhile, we establish the correlation between extremal graphs and spectral extremal graphs for $\mathcal{F}$.