In the current chemical literature, a large number of vertex–degree–based topological indices TI are considered, defined as the sum over all edges of the molecular graph of some function Ψ(x, y), where x and y are the degrees of the end-vertices of the respective edge. In order to find the minimal value of TI over benzenoid system sw ithh hexagons, we characterize convex benzenoid systems W such that ni(W )= ni(Sh), where ni is the number of internal vertices and Sh is the spiral benzenoid system .I f suchW does exist, then W has minimal TI -value. Otherwise, the spiral Sh has minimal TI -value.