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

Let \(\omega (G)\) be the number of components of graph G. For \(t\geqslant 0\) we call G t-tough if \(t\cdot \omega (G-X)\leqslant |X|\), for every \(X\subseteq V(G)\) with \(\omega (G-X)\geqslant 2\). \(1-\)tough graphs are also called Hamiltonian graphs. The eccentric connectivity index of a connected graph G denoted by \(\xi ^c(G)\), is defined as \(\xi ^c(G) = \sum _{v \in V(G)} \epsilon ({v}) d(v)\). The eccentric distance sum of a connected graph G is denoted by \(\xi ^d(G)\), is defined as \(\xi ^d(G) = \sum _{v \in V(G)} \epsilon (v) D(v)\). The connective eccentricity index of a connected graph G denoted as \(\xi ^{ce}(G)\), is defined as \(\xi ^{ce}(G) = \sum _{v \in V(G)} \frac{d(v)}{\epsilon (v)}\), where \(\epsilon (v)\) 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.