The sufficient conditions for $k$-leaf-connected graphs in terms of several topological indices
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| Format: | Preprint |
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2023
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| _version_ | 1866909162542202880 |
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| author | Ma, Tingyan Wang, Ligong Hu, Yang |
| author_facet | Ma, Tingyan Wang, Ligong Hu, Yang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_07093 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The sufficient conditions for $k$-leaf-connected graphs in terms of several topological indices Ma, Tingyan Wang, Ligong Hu, Yang Combinatorics 05C09, 05C35 G.2.2 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. |
| title | The sufficient conditions for $k$-leaf-connected graphs in terms of several topological indices |
| topic | Combinatorics 05C09, 05C35 G.2.2 |
| url | https://arxiv.org/abs/2304.07093 |