Related papers: Toughness and distance spectral radius in graphs i…

The concept of graph toughness was first introduced in 1973. In 1995, scholars first explored the lower bound of the toughness of connected d-regular graphs with respect to d and the second largest eigenvalue of the adjacency matrix. The…

The {\it toughness} $\tau(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a vertex cut in}~G\}$ for $G\ncong K_n,$ which was initially proposed by Chv\'{a}tal in 1973. A graph $G$ is called {\it $t$-tough} if $\tau(G)\geq t.$ Let…

Let G be a connected graph. The toughness of G is defined as t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of components of G-S.…

The Brouwer's toughness conjecture states that every $d$-regular connected graph always has $t(G)>\frac{d}{\lambda}-1$ where $\lambda$ is the second largest absolute eigenvalue of the adjacency matrix. In 1988, Enomoto introduced a…

Given a connected graph $G$, the toughness $\tau_G$ is defined as the minimum value of the ratio $|S|/\omega_{G-S}$, where $S$ ranges over all vertex cut sets of $G$, and $\omega_{G-S}$ is the number of connected components in the subgraph…

Let $G=(V(G),E(G))$ be a simple graph, where $V(G)$ and $E(G)$ are the vertex set and the edge set of $G$, respectively. The number of components of $G$ is denoted by $c(G)$. Let $t$ be a positive real number, and a connected graph $G$ is…

The scattering number $s(G)$ of graph $G=(V,E)$ is defined as $s(G)$=max\big\{$c(G-S)-|S|$\big\}, where the maximum is taken over all proper subsets $S\subseteq V(G)$, and $c(G-S)$ denotes the number of components of $G-S$. In 1988, Enomoto…

Let $t$ be a positive integer, and let $G$ be a connected graph of order $n$ with $n\geq t+2$. A graph $G$ is said to be $\frac{1}{t}$-tough if $|S|\geq\frac{1}{t}c(G-S)$ for every subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is…

The toughness of a graph $G$, denoted by $\tau(G)$, is defined by $\tau(G)=$min $\{\frac{|S|}{c(G-S)}:S\subseteq V(G)$ and $c(G-S)\geq2\}$. A graph $G$ is said to be $\tau$-tough if $\tau(G)\geq \tau$. Let $k\geq2$ be an integer. A tree $T$…

Let $\alpha\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=6$ for $\alpha\in[0,\frac{2}{3}]$ and $f(\alpha)=\frac{4}{1-\alpha}$ for $\alpha\in(\frac{2}{3},1)$. A graph $G$ is said to be…

The toughness $t(G)$ of a graph $G=(V,E)$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all $S\subset V$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. We…

Let $G$ be a graph. We denote by $c(G)$, $\alpha(G)$ and $q(G)$ the number of components, the independence number and the signless Laplacian spectral radius ($Q$-index for short) of $G$, respectively. The toughness of $G$ is defined by…

The study of the existence of hamiltonian cycles in a graph is a classic problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chv\'{a}tal's conjecture from another perspective: what is the spectral…

The toughness of a graph $G$ is defined as the largest real number $t$ such that for any set $S\subseteq V(G)$ such that $G-S$ is disconnected, $S$ has at least $t$ times more elements than $G-S$ has components (unless $G$ is complete, in…

The toughness of graph $G$, denoted by $\tau(G)$, is $\tau(G)=\min\{\frac{|S|}{c(G-S)}:S\subseteq V(G),c(G-S)\geq2\}$ for every vertex cut $S$ of $V(G)$ and the number of components of $G$ is denoted by $c(G)$. Bondy in 1973, suggested the…

A non-complete graph $G$ is said to be $t$-tough if for every vertex cut $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The toughness $\tau(G)$ of the graph $G$ is the maximum value of $t$ such that $G$…

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest…

The toughness $t(G)$ of a connected graph $G$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all proper subsets $S\subset V(G)$ such that $c(G-S)>1$, where $c(G-S)$ denotes the number of components of…

Let $G$ be a connected graph with $n$ vertices. The isolated toughness of $G$, denoted by $I(G)$, is defined by $I(G)=\min\left\{\frac{|S|}{i(G-S)}:S\subseteq V(G) \ \mbox{and} \ i(G-S)\geq2\right\}$ if $G$ is not complete, or…

In 1995, Brouwer proved that the toughness of a connected $k$-regular graph $G$ is at least $k/\lambda-2$, where $\lambda$ is the maximum absolute value of the non-trivial eigenvalues of $G$. Brouwer conjectured that one can improve this…