Vertex-count bounds for bounded scalar valence ($\frac{1}{2}V\le L-1+\frac{1}{2}n\le 2V$)

For a connected Feynman diagram with external valence $n$ and cubic, quartic and sextic vertices, counting half-edges gives $2I+n=3V_3+4V_4+6V_6$. Using $L=I-(V_3+V_4+V_6)+1$ yields $L-1+n/2=V_3/2+V_4+2V_6$. Since each summand lies between half and twice its vertex count, the displayed bounds follow. More generally vertices of valence between $r_{\min }$ and $r_{\max }$ give $(r_{\min }/2-1)V\le L-1+n/2\le (r_{\max }/2-1)V$.