# On Sum--Connectivity Index of Bicyclic Graphs

## TL;DR

This paper investigates the extremal sum--connectivity indices of bicyclic graphs with given vertices and matching number, identifying the graphs that maximize or minimize this index.

## Contribution

It determines the minimum, maximum, and second extremal sum--connectivity indices of bicyclic graphs for specified parameters and characterizes the extremal graphs.

## Key findings

- Identified graphs with minimum sum--connectivity index.
- Identified graphs with maximum sum--connectivity index.
- Characterized extremal graphs for given parameters.

## Abstract

We determine the minimum sum--connectivity index of bicyclic graphs with $n$ vertices and matching number $m$, where $2\le m\le \lfloor\frac{n}{2}\rfloor$, the minimum and the second minimum, as well as the maximum and the second maximum sum--connectivity indices of bicyclic graphs with $n\ge 5$ vertices. The extremal graphs are characterized.