Some New Results on the Sum of Squares of Eccentricity in Graphs

Abstract

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.