# Degenerate Tur\'an problems for hereditary properties

## TL;DR

This paper establishes bounds on the spectral radius of graphs that avoid certain subgraphs and induced subgraphs, extending previous results and providing new insights into the spectral properties of such graphs.

## Contribution

It proves that graphs with no copies of a fixed graph and no induced complete bipartite subgraph have spectral radius bounded by a function of their number of vertices, generalizing prior bounds.

## Key findings

- Spectral radius is bounded by O(n^{1-1/s}) for graphs avoiding H and induced K_{s,t}
- Extends previous bounds on spectral radius for graphs excluding certain subgraphs
- Provides a spectral perspective on Turán-type problems for hereditary properties

## Abstract

Let $H$ be a graph and $t\geq s\geq 2$ be integers. We prove that if $G$ is an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then $\lambda(G) = O\left(n^{1-1/s}\right)$ where $\lambda(G)$ is the spectral radius of the adjacency matrix of $G$. Our results are motivated by results of Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph with no copy (not necessarily induced) of $K_{s,t}$.