On sufficient condition for t-toughness of a graph in terms of eccentricity-based indices

Let \(\omega (G)\) $\omega (G)$ be the number of components of graph G. For \(t\geqslant 0\) $t⩾0$ we call G t-tough if \(t\cdot \omega (G-X)\leqslant |X|\) $t·\omega (G-X)⩽|X|$ , for every \(X\subseteq V(G)\) $X\subseteq V(G)$ with \(\omega (G-X)\geqslant 2\) $\omega (G-X)⩾2$ . \(1-\) $1-$ tough graphs are also called Hamiltonian graphs. The eccentric connectivity index of a connected graph G denoted by \(\xi ^c(G)\) ${\xi }^c(G)$ , is defined as \(\xi ^c(G) = \sum _{v \in V(G)} \epsilon ({v}) d(v)\) ${\xi }^c(G)={\sum }_{v\in V(G)}ϵ(v)d(v)$ . The eccentric distance sum of a connected graph G is denoted by \(\xi ^d(G)\) ${\xi }^d(G)$ , is defined as \(\xi ^d(G) = \sum _{v \in V(G)} \epsilon (v) D(v)\) ${\xi }^d(G)={\sum }_{v\in V(G)}ϵ(v)D(v)$ . The connective eccentricity index of a connected graph G denoted as \(\xi ^{ce}(G)\) ${\xi }^{\mathit{ce}}(G)$ , is defined as \(\xi ^{ce}(G) = \sum _{v \in V(G)} \frac{d(v)}{\epsilon (v)}\) ${\xi }^{\mathit{ce}}(G)={\sum }_{v\in V(G)}\frac{d(v)}{ϵ(v)}$ , where \(\epsilon (v)\) $ϵ(v)$ is the eccentricity of the vertex v, D(v) is the sum of the distance from to all other vertices, and d(v) is the degree of vertex v. Finding sufficient conditions for a graph to possess certain properties is a meaningful and important problem. In this article, we give sufficient conditions for t-toughness graphs in terms of the eccentric connectivity index, eccentric distance sum, and connective eccentricity index.