# The Extremal Function and Colin de Verdi\`{e}re Graph Parameter

## TL;DR

This paper investigates the maximum edges in graphs constrained by the Colin de Verdière parameter, proposing a conjecture and proving it for specific graph classes, linking it to the graph complement conjecture.

## Contribution

The paper introduces a conjecture on edge bounds related to the Colin de Verdière parameter and proves it for certain classes of graphs, advancing understanding of spectral graph parameters.

## Key findings

- Proposed a conjecture relating edges and the Colin de Verdière parameter.
- Proved the conjecture for graphs with μ(G) ≤ 7.
- Proved the conjecture for graphs with μ(G) ≥ |V(G)|-6 and for chordal graphs.

## Abstract

We study the maximum number of edges in an $n$ vertex graph with Colin de Verdi\`{e}re parameter no more than $t$. We conjecture that for every integer $t$, if $G$ is a graph with at least $t$ vertices and Colin de Verdi\`{e}re parameter at most $t$, then $|E(G)| \leq t|V(G)|-\binom{t+1}{2}$. We observe a relation to the graph complement conjecture for the Colin de Verdi\`{e}re parameter and prove the conjectured edge upper bound for graphs $G$ such that either $\mu(G) \leq 7$, or $\mu(G) \geq |V(G)|-6$, or the complement of $G$ is chordal, or $G$ is chordal.