In 2010, Vukičević and Gašperov proposed a group of 148 discrete Adriatic indices which have represented good predictive characteristics on the testing sets conducted by the International Academy of Mathematical Chemistry (IAMC). One of such indices was the misbalance Hadeg ( \(\mathcal{M}\mathcal{H}\) ) index which showed good ability in predicting the enthalpy of vaporization and standard enthalpy of vaporization for octane isomers. For a simple graph G with the edge set \(E_G\) , the \(\mathcal{M}\mathcal{H}\) index is expressed by \(\mathcal{M}\mathcal{H}(G)=\sum _{\alpha \beta \in E_G} \Big |\Big (\frac{1}{2}\Big )^{d^{G}_{\alpha }}-\Big (\frac{1}{2}\Big )^{d^{G}_{\beta }}\Big |\) , where \(d^{G}_{\beta }\) stands for the degree of the vertex \(\beta \) in G. Despite passing almost 14 years since the introduction of the \(\mathcal{M}\mathcal{H}\) index, it has been relatively less investigated and left into oblivion. A main purpose of this research is to attract the attention of the mathematical chemistry community towards this molecular structure descriptor by studying some of its basic mathematical properties over trees, molecular trees, unicyclic graphs, bicyclic graphs and, in general, over c-cyclic graphs. More specifically, we give the minimum values of the \(\mathcal{M}\mathcal{H}\) index over trees, unicyclic graphs and bicyclic graphs of a given number of vertices and with a fixed maximum degree and characterize the corresponding minimal graphs. The special case when the class of considered trees is restricted to the class of molecular trees and the general case when the class of considered unicyclic and bicyclic graphs is extended to the class of c-cyclic graphs are also examined.
