Harmonic-Arithmetic Index: Lower Bound for n-Order Trees With Fixed Number of Pendant Vertices and Monogenic Semigroup for Graph Operations

Graph theory combined with chemistry provides a strong framework for modeling and assessing chemical compounds. By representing molecules as graphs and applying topological indices, chemists can gain profound insights into the physical and chemical characteristics of compounds. This approach not only deepens theoretical understanding but also enhances practical applications in drug development, materials science, and chemical research, leading to more effective and innovative solutions in these fields. Topological indices are valuable tools for chemists to predict and analyze the physical and chemical properties of molecules based on their structural characteristics. By quantifying aspects such as branching, connectivity, and overall size, these indices provide insights into boiling and melting points, stability, reactivity, and other crucial properties. This correlation helps in designing and understanding molecules with desired characteristics for various applications in chemistry and materials science as well. Among the various degree-based topological indices, our focus is on the Harmonic-Arithmetic (HA) index. This newly introduced degree-based topological index is designed to offer a novel perspective on analyzing molecular structures. By integrating principles from both harmonic and arithmetic means, it offers a refined measure of molecular connectivity and complexity. The HA index of a graph G is defined as HAG=∑uv∈EG4duuv/du+dv2, where du and dv are denoted as degrees of nodes u and v. This paper provides the complete solution of minimal version of HA index of a tree Sn,l with n vertices and l leaves. In addition, the HA index of two monogenic semigroup graphs by Cartesian product and tensor product is calculated.