A general theorem in spectral extremal graph theory

TL;DR

This paper establishes a general theorem in spectral extremal graph theory that characterizes spectral extremal graphs for various forbidden subgraph families, unifying and extending previous results.

Contribution

It provides a broad characterization theorem linking extremal and spectral extremal graphs for many families, including those containing complete bipartite graphs.

Findings

Spectral extremal graphs often contain the same large bipartite subgraphs as extremal graphs.

The theorem applies to a wide range of forbidden families, unifying multiple results.

A relation between spectral extremal graphs and those maximizing the spectral radius of $A_\alpha$ is established.

Abstract

The extremal graphs $\mathrm{EX}(n,\mathcal F)$ and spectral extremal graphs $\mathrm{SPEX}(n,\mathcal F)$ are the sets of graphs on $n$ vertices with maximum number of edges and maximum spectral radius, respectively, with no subgraph in $\mathcal F$. We prove a general theorem which allows us to characterize the spectral extremal graphs for a wide range of forbidden families $\mathcal F$ and implies several new and existing results. In particular, whenever $\mathrm{EX}(n,\mathcal F)$ contains the complete bipartite graph $K_{k,n-k}$ (or certain similar graphs) then $\mathrm{SPEX}(n,\mathcal F)$ contains the same graph when $n$ is sufficiently large. We prove a similar theorem which relates $\mathrm{SPEX}(n,\mathcal F)$ and $\mathrm{SPEX}_\alpha(n,\mathcal F)$, the set of $\mathcal F$-free graphs which maximize the spectral radius of the matrix $A_\alpha=\alpha D+(1-\alpha)A$, where $A$ is the adjacency matrix and $D$ is the diagonal degree matrix.