Complete solution to open problems on exponential augmented Zagreb index of chemical trees

Abstract

One of the crucial problems in combinatorics and graph theory is characterizing extremal structures with respect to graph invariants from the family of chemical trees. Cruz et al. (2020) [7] presented a unified approach to identify extremal chemical trees for degree-based graph invariants in terms of graph order. The exponential augmented Zagreb index (EAZ) is a well-established graph invariant formulated for a graph G asEAZ(G)=∑vivj∈E(G)e(didjdi+dj−2)3, where di signifies the degree of vertex vi, and E(G) is the edge set. Due to some special counting features of EAZ, it was not covered by the aforementioned unified approach. As a result, the exploration of extremal chemical trees for this invariant was posed as an open problem in the same article. The present work focuses on generating a complete solution to this problem. Our findings offer maximal and minimal chemical trees of EAZ in terms of the graph order n.