New Relations between Zagreb Indices and Omega Invariant

In this work, we studied the problem of determining the values of the Zagreb indices of all the realizations of a given degree sequence. Methods: We first obtained some new relations between the first and second Zagreb indices and the forgotten index sometimes called the third Zagreb index. These relations also include the triangular numbers, order, size, and the biggest vertex degree of a given graph. As the first Zagreb index and the forgotten index of all the realizations of a given degree sequence are fixed, we concentrated on the values of the second Zagreb index and studied several properties including the effect of vertex addition. Results: In our calculations, we make use of a new graph invariant, called omega invariant, to reach numerical and topological values claimed in the theorems. This invariant is closely related to Euler char-acteristic and the cyclomatic number of graphs. Conclusion: Therefore this invariant is used in the calculation of some parameters of the molecular structure under review in terms of vertex degrees, eccentricity, and distance.