Compact valence sequences for molecules with single, double and triple covalent bonds. II. Graphs with non-trivial cycles

Molecular graphs able to model covalent multiple bonds are called plerographs. For such graphs, with single, double and triple edges, compact degree sequences (n(1), n(2), n(3), n(4)), where n(i), i = 1, 2, 3, 4 denote the number of vertices of degree i, are defined. Investigations are extended to necessary and sufficient conditions on (n(1), n(2), n(3), n(4)) for the existence of graph G such that G has n(1), n(2), n(3) and n(4) vertices of degrees 1, 2, 3 and 4. Namely, graphs with non-trivial cycles with at least three vertices are allowed.