The sum of the squares of eccentricity \((SSE)\) over all vertices of a connected graph is a new graph invariant proposed in \([13]\) and further studied in \([14, 15]\). In this paper, we report some further mathematical properties of \(SSE\). We give sharp lower bounds for \(SSE\) among all \(n\)-vertices connected graphs with given independence number, vertex-, and edge-connectivity, respectively. Addtionally, we give explicit formulas for \(SSE\) of Cartesian product of two graphs, from which we deduce \(SSE\) of \(C_4\), nanotube and nanotorus.
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