Strict unimodality without log-concavity for partial Petrial polynomials

For a simple graph G, the partial Petrial polynomial PG×(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 Rₘ 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 PRₘ×(z) is strictly unimodal for every m≥1, but is log-concave if and only if 1≤ m≤3. Thus the $21$-vertex tree R₄ provides an explicit strictly unimodal, non-log-concave example. For m≥5, the unique mode occurs at degree (149m+85)/32. 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.