The Extremal Function and Colin de Verdière Graph Parameter

Abstract

We study the maximum number of edges in an $n$ vertex graph with Colin de Verdière 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 Verdière parameter at most $t$, then $|E(G)|\le t|V(G)|-\left(\genfrac{}{}{0ex}{}{t+1}{2}\right)$. We observe a relation to the graph complement conjecture for the Colin de Verdière parameter and prove the conjectured edge upper bound for graphs $G$ such that either $\hat{I}¼(G)\le 7$, or $\hat{I}¼(G)\ge |V(G)|-6$, or the complement of $G$ is chordal, or $G$ is chordal.

11 pages