Harmonic-Arithmetic Index of Unicyclic Graphs

Document Type : Research Paper

Authors

School of Mathematics‎, ‎Nanjing University of Aeronautics and Astronautics‎, ‎Nanjing‎, ‎China

10.22052/ijmc.2024.255486.1909

Abstract

‎Let G=(V(G),E(G)) be a graph‎. ‎The harmonic-arithmetic index of G is defined as $HA(G)=\sum_{uv\in E(G)}\frac{4d_{G}(u)d_{G}(v)}{(d_{G}(u)+d_{G}(v))^2}$‎, ‎where $d_{G}(u)$ is the degree of a vertex $u\in V(G)$‎. ‎In this paper‎, ‎we consider the upper and lower bounds of the harmonic-arithmetic index of unicyclic graphs with a fixed order‎. ‎Furthermore‎, ‎the graphs attaining the extremal values are also characterised.

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