ScholarWorks@성균관대학교

초록

The exponential augmented Zagreb ( EAZ ) index is a graph-theoretical descriptor that correlates strongly with the physico-chemical properties of molecules. Introduced by Rada in 2019, it is defined for a simple graph ϒ as EAZ(ϒ)=∑vivj∈E(ϒ)e(didjdi+dj−2)3, where E(ϒ) denotes the edge set and di is the degree of vertex vi. This work is motivated by some open problems concerning the well-known augmented Zagreb index ( AZ ). In particular, the maximization of AZ for a given graph order and a specified number of pendant vertices was posed as an open problem in Chen et al. (2022) [7] . We completely resolve this problem for the exponential version, EAZ . In recent work Xu et al. (2025) [38] , two related questions were raised: whether the maximal graphs for AZ and EAZ coincide, and if not, how they differ. We provide complete answers to these questions with respect to the chromatic number and the number of pendant vertices. We explore the maximal graph for EAZ in terms of chromatic number and graph order, and show that this differs substantially from the corresponding extremal graph for AZ . Further results include a characterization of the maximal graphs for EAZ when vertex connectivity and edge connectivity are prescribed together with the graph order. In addition, we prove that EAZ(ϒ) increases upon adding an edge to ϒ, a crucial result for understanding the extremal properties of EAZ . Finally, the potential usefulness of this discrete invariant in chemical graph theory is demonstrated.

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