# First and Second Maximum of Randi\'{c} Index Among all $k-$Cyclic Graphs of a Given Order

## TL;DR

This paper determines the graphs with the highest and second-highest Randić index among all k-cyclic graphs with a fixed number of vertices, advancing understanding of extremal properties in graph theory.

## Contribution

It explicitly computes the first and second maximum Randić indices among all n-vertex k-cyclic graphs, filling a gap in extremal graph theory.

## Key findings

- Identified the graphs with maximum Randić index for given n and k.
- Determined the second maximum Randić index for these graphs.

## Abstract

Suppose $G$ is a simple graph with edge set $E(G)$. The Randi\'{c} index $R(G)$ is defined as $R(G)=\sum_{uv\in E(G)}\frac{1}{\sqrt{deg_{G}(u)deg_{G}(v)}}$, where $deg_G(u)$ denotes the vertex degree of $u$ in $G$. In this paper, the first and second maximum of Randi\'{c} index among all $n-$vertex $k-$cyclic graphs were computed.