Some lower and upper bounds on the third ABC index

Atom-bond connectivity (ABC) index has been applied up to now to study the stability of alkanes and the strain energy of cycloalkanes. Graovac defined the second ABC index as ABC2(G)=∑vivj∈E(G)1ni+1nj−2ninj, and Kinkar studied the upper bounds. In this paper, we define a new index which is called the third ABC index and it is defined as equation(1)ABC3(G)=∑vivj∈E(G)1ei+1ej−2eiej, and we present some lower and upper bounds on ABC3 index of graphs.