Harmonic-Arithmetic Index of Unicyclic Graphs with given Girth and Connected Graphs with Minimum Degree
نویسندگان
1 Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Tamil Nadu 603203, India
2 Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Tamil Nadu 603203, India
3 Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Tamil Nadu 603203, India
4 Department of Mathematics, School of Arts, Sciences, Humanities, and Education, SASTRA Deemed University, Thanjavur 613401, India
doi
10.22052/ijmc.2025.257071.2022چکیده
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.
