University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 17 1 2026 03 01 Harmonic-Arithmetic‎ ‎Index‎ ‎of‎ ‎Unicyclic‎ ‎Graphs‎ ‎with given Girth and Connected Graphs with Minimum Degree 35 51 115415 10.22052/ijmc.2025.257071.2022 EN Jayavel Pooja Department of Mathematics‎, ‎College of Engineering and Technology‎, ‎Faculty of Engineering and Technology‎, ‎SRM Institute of‎ ‎Science and Technology‎, ‎Kattankulathur‎, ‎Tamil Nadu 603203‎, ‎India ‎Venkatesan Muthukumaran Department of Mathematics‎, ‎College of Engineering and Technology‎, ‎Faculty of Engineering and Technology‎, ‎SRM Institute of‎ ‎Science and Technology‎, ‎Kattankulathur‎, ‎Tamil Nadu 603203‎, ‎India ‎Suresh Elumalai Department of Mathematics‎, ‎College of Engineering and Technology‎, ‎Faculty of Engineering and Technology‎, ‎SRM Institute of‎ ‎Science and Technology‎, ‎Kattankulathur‎, ‎Tamil Nadu 603203‎, ‎India Selvaraj Balachandran Department of Mathematics, School of Arts, Sciences, Humanities, and Education, SASTRA Deemed University, Thanjavur 613401, India Journal Article 2025 06 18 ‎Let G be the finite‎, ‎simple‎, ‎and connected graph with a vertex set as V(G) and an edge set as E(G)‎. ‎The harmonic-arithmetic index of graph G is defined as $HA(G) = \sum\limits_{\rho\phi \in E(G)} {\dfrac{{4{d_\rho}{d_\phi}}}{{{{({d_\rho}‎ + ‎{d_\phi})}^2}}}}$ where $d_\rho$ denotes the degree of the vertex $\rho$ and $\rho\phi$ denotes the edge‎. ‎Let $U_{\eta,\mathfrak{g}}$ be the set of unicyclic graphs with $\eta$ vertices and given girth g‎. ‎Let $G_{\eta,\delta}$ be the set of simple connected graphs with $\eta$ vertices with minimum degree $\delta$‎. ‎In this article‎, ‎we present the maximum and second-maximum harmonic-arithmetic index of unicyclic graphs with a given girth and determine their corresponding graphs‎. ‎The obtained results remain valid when the analysis is confined to the class of chemical unicyclic graphs‎. ‎Further‎, ‎we obtain extremal graphs in $G_{\eta‎, ‎\delta}$ for which the HA index reaches its smallest value‎, ‎or we provide a lower bound‎, ‎for $\delta \geq\left\lceil \delta_0 \right\rceil$‎, ‎with $\delta_0 = p_0(\eta-1)$‎, ‎where $p_0 \approx 0.23606$ is the distinct positive root of the expression p^2‎ + ‎4p‎ -‎1 =0‎. ‎We demonstrate that the extremal graphs are regular graphs of degree $\delta$ when $\delta$ or $\eta$ is even‎. https://ijmc.kashanu.ac.ir/article_115415_39336345e5150c0ee6b45f519c13bbd5.pdf