Resolving Open Problems on the Hyper-Zagreb Index and its Chemical Applications

TL;DR

This paper resolves open problems regarding bounds on the hyper-Zagreb index in graphs with fixed connectivity, identifies extremal graphs, and explores chemical applications through QSPR studies.

Contribution

It determines extremal graphs that maximize the hyper-Zagreb index under connectivity constraints and extends analysis to other graph parameters, with chemical relevance.

Findings

Identified extremal graphs for maximum hyper-Zagreb index under fixed connectivity.

Extended extremal analysis to graphs with specified leaves, chromatic number, and independence number.

Demonstrated chemical application of the hyper-Zagreb index through QSPR studies.

Abstract

Topological indices are numerical invariants derived from molecular graphs and play an important role in characterizing chemical compounds and predicting their properties. Among the earliest descriptors are the classical Zagreb indices introduced by Gutman and Trinajsti\'c in 1972. A more recent development is the hyper-Zagreb index ($HM$), defined as $HM(G)=\sum_{v_i v_j\in E(G)}(d_i+d_j)^2$, where $d_i$ denotes the degree of vertex $v_i$. In 2023, Hayat et al. posed an open problem concerning bounds on the $HM$ index under fixed vertex-connectivity or edge-connectivity, along with the characterization of the corresponding extremal graphs. In this work, the problem is resolved by determining the extremal graphs that maximize $HM$ index under these constraints. The investigation is further extended to several additional extremal problems, including graphs with a given number of leaves, chromatic number, and independence number. The associated extremal graphs are identified in each case. In addition, the chemical relevance of $HM$ is examined through QSPR studies. Finally, the conclusion is presented.