The sum of squares of eccentricity of connected graphs

The sum of the squares of eccentricity (SSE) over all vertices of a connected graph is also called the first normalized Zagreb eccentricity index of this graph in [18]. In this paper, we report some new mathematical properties of SSE. We first give upper and lower bounds for SSE among n-vertices connected graphs. Then we give a lower bound for SSE in terms of Wiener index and an upper bound for SSE in terms of the first Zagreb index among n-vertices connected graphs. Finally, we give a Nordhaus-Gaddum-type bound for SSE.