Strict unimodality without log-concavity for partial Petrial polynomials

Abstract

For a simple graph $G$, the partial Petrial polynomial $P_G^{\times}(z)$ enumerates all diagonal perturbations of its adjacency matrix over $GF(2)$ by rank. We derive a rooted attachment identity and apply it to the tree $R_m$ obtained by joining a new vertex to the middle vertex of each of $m$ disjoint copies of the five-vertex path. We prove that the nonzero coefficient sequence of $P_{R_m}^{\times}(z)$ is strictly unimodal for every $m\ge1$, but is log-concave if and only if $1\le m\le3$. Thus the $21$-vertex tree $R_4$ provides an explicit strictly unimodal, non-log-concave example. For $m\ge5$, the unique mode occurs at degree $\lfloor(149m+85)/32\rfloor$. More generally, for any fixed path rooted at an internal vertex, joining a new vertex to the roots of sufficiently many disjoint copies produces a tree whose partial Petrial polynomial is not log-concave.