THE SECOND ZAGREB SPECTRAL RADIUS OF UNICYCLIC GRAPHS

For a connected simple unicyclic graph on n ≥ 5 vertices,
we determine the maximum spectral radius of the second Zagreb matrix
and characterize equality. Except at n = 6, the unique extremal graph
is a triangle with its n − 3 remaining vertices attached as leaves as
evenly as possible to two vertices of the triangle. At n = 6, the unique
extremal graph has one leaf at each triangle vertex. This establishes the
conjectured extremizers of Jiang, Zuo, Wang, and Jahanbani for every
order. The proof combines a previously established structural reduction
with a Schur-complement spectral comparison and eight finite, exact
integer-polynomial positivity certificates. The latter are supplied with a
reproducible standalone verifier.