Linear spectral Turan problems for expansions of graphs with given chromatic number

TL;DR

This paper establishes sharp bounds on the maximum edges and spectral radius of linear hypergraphs that avoid certain expanded graphs with given chromatic number, linking spectral properties to graph expansions.

Contribution

It introduces new bounds for linear hypergraph extremal problems by connecting spectral radii to shadow graphs for graphs with specific chromatic properties.

Findings

Sharp bounds for maximum edges in linear hypergraphs avoiding certain expansions.

Asymptotic bounds for spectral radius of such hypergraphs.

Connection between spectral radii of hypergraphs and their shadow graphs.

Abstract

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. The $r$-expansion $F^{r}$ of a graph $F$ is the $r$-uniform hypergraph obtained from $F$ by enlarging each edge of $F$ with a vertex subset of size $r-2$ disjoint from the vertex set of $F$ such that distinct edges are enlarged by disjoint subsets. Let $ex_{r}^{lin}(n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ be the maximum number of edges and the maximum spectral radius of all $F^{r}$-free linear $r$-uniform hypergraphs with $n$ vertices, respectively. In this paper, we present the sharp (or asymptotic) bounds of $ex_{r}^{lin}( n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ by establishing the connection between the spectral radii of linear hypergraphs and those of their shadow graphs, where $F$ is a $(k+1)$-color critical graph or a graph with chromatic number $k$.