A flower graph consists of a half line and $N$ symmetric loops connected at a
single vertex with $N \geq 2$ (it is called the tadpole graph if $N = 1$). We
consider positive single-lobe states on the flower graph in the framework of
the cubic nonlinear Schrodinger equation. The main novelty of our paper is a
rigorous application of the period function for second-order differential
equations towards understanding the symmetries and bifurcations of standing
waves on metric graphs. We show that the positive single-lobe symmetric state
(which is the ground state of energy for small fixed mass) undergoes exactly
one bifurcation for larger mass, at which point $(N-1)$ branches of other
positive single-lobe states appear: each branch has $K$ larger components and
$(N-K)$ smaller components, where $1 \leq K \leq N-1$. We show that only the
branch with $K = 1$ represents a local minimizer of energy for large fixed
mass, however, the ground state of energy is not attained for large fixed mass.
Analytical results obtained from the period function are illustrated
numerically.