Turán coefficients of signed graphs

Source abstract

We use spectral methods to determine Turán coefficients of signed graphs under containment that preserves negative edges and allows positive edges to be realized by either sign. For a fixed forbidden signed graph $\sF$ with at least one edge, the coefficient $\tht(\sF)$ is the supremum of the weights $ρ\ge1$ for which every $\sF$-free signed graph $\sG$ of order $n$ satisfies $e^+(\sG)+ρe^-(\sG)\le\binom n2+o(n^2)$. We prove a variational characterization of the weighted extremal density and show that $\tht(\sF)=1+1/L(\sF)$, where $L(\sF)$ is the supremum of the adjacency spectral radii of $\sF$-admissible graphs. The coefficient is infinite exactly when the underlying graph $F$ is bipartite, and equals one exactly when $F$ has no independent set meeting every negative edge. When $χ(F)=3$, the possible coefficients are exactly $1$, $3/2$, $1+1/\sqrt3$, $(1+\sqrt5)/2$, $1+1/\sqrt2$, and $2$, each of which is realized by a signed graph. In this case, if $\tht(\sF)>1$, then $L(\sF)$ is attained by an admissible graph on at most five vertices, and the order bound is sharp. More generally, when some path is inadmissible, we obtain explicit order bounds for connected admissible graphs with bounded maximum degree. We also derive restrictions on the algebraic degree and rational values of $\tht(\sF)$.