Toughness, fractional extendability and distance spectral radius in graphs
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A graph $G$ is said to be $t$-tough if $|S|\geq t\cdot c(G-S)$ for every subset $S\subseteq V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ denotes the number of connected components in $G-S$. A graph $G$ is fractional $k$-extendable if every $k$-matching in $G$ can be extended to a fractional perfect matching of $G$. In this paper, we first establish an upper bound on the distance spectral radius of $G$ to ensure that $G$ is a $\frac{1}{t}$-tough graph. Then we give an upper bound on the distance spectral radius of $G$ to guarantee that $G$ is a $t$-tough graph. Finally, we show an upper bound on the distance spectral radius of $G$ to guarantee that $G$ is a fractional $k$-extendable graph.
Abstract
A graph $G$ is said to be $t$-tough if $|S|\geq t\cdot c(G-S)$ for every subset $S\subseteq V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ denotes the number of connected components in $G-S$. A graph $G$ is fractional $k$-extendable if every $k$-matching in $G$ can be extended to a fractional perfect matching of $G$. In this paper, we first establish an upper bound on the distance spectral radius of $G$ to ensure that $G$ is a $\frac{1}{t}$-tough graph. Then we give an upper bound on the distance spectral radius of $G$ to guarantee that $G$ is a $t$-tough graph. Finally, we show an upper bound on the distance spectral radius of $G$ to guarantee that $G$ is a fractional $k$-extendable graph.