The sufficient conditions for $k$-leaf-connected graphs in terms of several topological indices

TL;DR

This paper establishes new sufficient conditions for a graph to be $k$-leaf-connected using various topological indices, extending known results from Hamilton-connected graphs to broader classes.

Contribution

It introduces novel criteria based on Zagreb and hyper-Zagreb indices for determining $k$-leaf-connectedness, including conditions involving the complement graph.

Findings

Conditions involving Zagreb index for $k$-leaf-connectedness

Conditions involving reciprocal degree distance and hyper-Zagreb index

Sufficient criteria using indices of the complement graph

Abstract

Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $k\geq2$ and given any subset $S\subseteq|V(G)|$ with $|S|=k$, if a graph $G$ of order $|V(G)|\geq k+1$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T$, then the graph $G$ is a $k$-leaf-connected graph. A graph $G$ is called Hamilton-connected if any two vertices of $G$ are connected by a Hamilton path. Based on the definitions of $k$-leaf-connected and Hamilton-connected, we known that a graph is $2$-leaf-connected if and only if it is Hamilton-connected. During the past decades, there have been many results of sufficient conditions for Hamilton-connected with respect to topological indices. In this paper, we present sufficient conditions for a graph $G$ to be $k$-leaf-connected in terms of the Zagreb index, the reciprocal degree distance or the hyper-Zagreb index. Furthermore, we use the first Zagreb index and hyper-Zagreb index of the complement graph $\overline{G}$ to give sufficient conditions for a graph $G$ to be $k$-leaf-connected.