On the maximum square PI index of bicyclic graphs

The square PI index of a connected graph sums, over its edges, the squares of the numbers of verticescloser to each endpoint. We prove that among all connected bicyclic graphs of any order at least five,the unique maximizer is obtained from the complete bipartite graph with parts of sizes two and three byattaching every remaining vertex as a leaf to a degree-two vertex. This settles the conjecture of Alex, Gutman,Heidari and Rostami and extends its proposed order range. The proof combines a bridge contraction, strictconvexity of leaf allocations, a uniform bound for nonbridge edges, and a finite classification of bicycliccores.