Algebraic connectivity of connected graphs with fixed number of pendant vertices

TL;DR

This paper investigates the extremal algebraic connectivity in connected graphs with a fixed number of pendant vertices and explores related properties in unicyclic graphs.

Contribution

It characterizes the graphs that maximize or minimize algebraic connectivity among all connected graphs with given pendant vertices and discusses algebraic connectivity in unicyclic graphs.

Findings

Identifies extremal graphs for algebraic connectivity with fixed pendant vertices

Provides bounds and properties for algebraic connectivity in unicyclic graphs

Enhances understanding of spectral graph theory in constrained graph classes

Abstract

In this paper we consider the following problem: Over the class of all simple connected graphs of order $n$ with $k$ pendant vertices ($n,k$ being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.