Graphs with large rank numbers and rank numbers of subdivided stars

Figures & data

Fig. 1 $S_{9,6,5,3,2}$ with corresponding matrix $C$.

Fig. 2 $S=S^1=S_{32,25,3,2}$ with corresponding matrix $C=C^1$.

Fig. 3 $S^2=S_{25,3,2}$ with corresponding matrix $C^2$.

Fig. 4 $S^3=S_{9,3,2}$ with corresponding matrix $C^3$.

Fig. 5 $S^4=S_{3,2,1}$ with corresponding matrix $C^4$.

Fig. 6 ${\chi }_r\left(S_{32,25,3,2}\right)={\chi }_r\left(P_{32}\right)=6$.

Fig. 7 $J$ must contain one of these four graphs.