The ABC Matrix

The atom-bond connectivity index—abbreviated ABC—has proven to describe the heat of formation of alkanes with an accuracy comparable to that of high-level ab initio and DFT methods. Its physical interpretation in terms of quantum-theory, its generalization, and many of its mathematical properties have been reported. Here, a probabilistic interpretation of the generalized ABC index is provided. It indicates that the terms defining this index represent the probability of visiting a nearest neighbor edge from one side or the other of a given edge in a graph. This interpretation is general enough as to cover the case of polarizing bonds in a molecular context. We then introduce a matrix representation of these probabilities in the form of generalized ABC matrices. Using this representation we study some indices related to the eigenvalues of the ABC matrices, such as the ABC energy and the ABC Estrada index. We provide some bounds for these parameters in general graphs.