# Inequalities for doubly nonnegative functions

## TL;DR

This paper investigates inequalities involving doubly nonnegative functions and graph homomorphism densities, conjecturing a lower bound related to the spectral properties of the functions, and proves it for several classes of graphs.

## Contribution

The paper introduces a conjecture connecting graph homomorphism densities with spectral properties of functions and proves it for specific graph classes.

## Key findings

- Proved the conjecture for complete graphs.
- Established the inequality for unicyclic and bicyclic graphs.
- Validated the conjecture for all graphs with up to 5 vertices.

## Abstract

Let $g$ be a bounded symmetric measurable nonnegative function on $[0,1]^2$, and $\left\lVert g \right\rVert = \int_{[0,1]^2} g(x,y) dx dy$. For a graph $G$ with vertices $\{v_1,v_2,\ldots,v_n\}$ and edge set $E(G)$, we define \[ t(G,g) \; = \; \int_{[0,1]^n} \prod_{\{v_i,v_j\} \in E(G)} g(x_i,x_j) \: dx_1 dx_2 \cdots dx_n \; . \] We conjecture that $t(G,g) \geq \left\lVert g \right\rVert^{|E(G)|}$ holds for any graph $G$ and any function $g$ with nonnegative spectrum. We prove this conjecture for various graphs $G$, including complete graphs, unicyclic and bicyclic graphs, as well as graphs with $5$ vertices or less.