The minimal size of graphs with given pendant-tree connectivity

TL;DR

This paper investigates the minimal number of edges required in graphs of a given size to achieve specific pendant-tree connectivity levels, providing exact values and bounds for this parameter.

Contribution

It introduces precise bounds and exact values for the minimal edge count in graphs with specified pendant-tree connectivity, advancing understanding of graph connectivity measures.

Findings

Derived exact values for minimal edges in certain graphs

Established sharp bounds for the parameter $f(n,k, au_k)$

Extended classical connectivity concepts to pendant-tree connectivity

Abstract

The concept of pendant-tree $k$-connectivity $\tau_k(G)$ of a graph $G$, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. Let $f(n,k,\ell)$ be the minimal number of edges of a graph $G$ of order $n$ with $\tau_k(G)=\ell \ (1\leq \ell\leq n-k)$. In this paper, we give some exact value or sharp bounds of the parameter $f(n,k,\ell)$.