  The total eccentricity index of a connected graph is defined as sum of the
eccentricities of all its vertices. We denote the set of all connected graphs
on $n$ vertices with $k$ pendant vertices by $\mathfrak{H}_{n,k}$ and denote
the set of all connected graphs on $n$ vertices with $s$ cut vertices by
$\mathfrak{C_{n,s}}$. In this paper, we give the sharp lower and upper bounds
on the total eccentricity index over $\mathfrak{H}_{n,k}$ and the sharp lower
bound for the same over $\mathfrak{C_{n,s}}$. We also provide the sharp upper
bounds on the total eccentricity index over $\mathfrak{C_{n,s}}$ when
$s=0,1,n-3,n-2$ and propose a problem regarding the upper bound over
$\mathfrak{C_{n,s}}$ for $2\leq s\leq n-4.$
