On acyclic molecular graphs with prescribed numbers of edges that connect vertices with given degrees

We find a necessary and sufficient conditions on a sequence (m(11), m(12) ,m(13), m(14), m(22), m(23), m(24), m(33), m(34), m(44)) for the existence of an acyclic molecular graph G such that exactly m(ij) edges connect vertices of degree i and j. We use this result together with two additional results to make an algorithm that generates all the sequences (m(11), m(12), m(13), m(14), m(22), m(23), m(24), m(33), m(34), m(44)) such that a molecular acyclic graph exists with exactly m(ij) edges connecting vertices of degree i and j. This algorithm is utilized to compare discriminative properties of the Zagreb index and the modified Zagreb index, and it is found that the modified Zagreb index is more discriminative then the Zagreb index.