Compact valence sequences for molecules with single, double and triple covalent bonds

Molecular graphs able to model covalent multiple bonds are called plerographs. For such graphs with single, double and triple edges, we define here the compact degree sequence (n(1), n(2), n(3), n(4)), where n(i), i = 1, 2, 3, 4 denote the number of vertices of degree i. We have found the necessary and sufficient conditions (n(1), n(2), n(3), n(4)) for the existence of graph G such that G has nj, n2, n3 and n4 vertices of degrees 1, 2, 3 and 4, and have formulated the findings into three theorems