We have considered the combinatorics of the Petersen graph and its various compositions with other graphs for all irreducible representations (IRs) of the corresponding automorphism groups. These graphs represent Berry pseudorotation of the trigonal bipyrmidal (TBP) molecules, clusters such as $$\hbox {Ta}_{5}$$ Ta 5 , and non-rigid ligands placed at the corners of the TBP structure. We have computed the properties for coloring combinatorics for the vertices, edges, and pentagons of the Petersen graph for all IRs for up to six different color types. Compositions of the Petersen graphs with other graphs have also been considered with complete enumeration tables obtained for the composition of the Petersen graph with $$\hbox {K}_{2}$$ K 2 containing 122,880 elements and 136 conjugacy classes. Such compositions of the Petersen graph represent the automorphism groups of the isomerization graphs of pentacoordinated TBP molecules and transition metal clusters such as $$[\hbox {Co}(\hbox {H}_{2}\hbox {O})_{5}]^{3+}$$ [ Co ( H 2 O ) 5 ] 3 + , pentacoordinated transition metal-ammonia complexes, organometallic systems such as $$\hbox {Sb}(\hbox {CH}_{3})_{5}$$ Sb ( CH 3 ) 5 etc., all of which exhibit isomerization graphs that have wreath product automorphisms. Coloring tables of such graphs under their group actions are also constructed.