Walks, Paths and Connectivity · Hard problem

A wheel graph has a hub vertex $0$ joined to every vertex of a $5$-cycle $1\text{-}2\text{-}3\text{-}4\text{-}5\text{-}1$ (edges $12,23,34,45,51,01,02,03,04,05$). A second, separate graph has two triangles $a,b,c$ and $d,e,f$, joined by a single bridge $cd$ (edges $ab,bc,ca,de,ef,fd,cd$). How many MORE triangles does the wheel graph have than the second graph?