On the sum of squares of eccentricity

Abstract

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 [15]. In this paper, we explore some new properties of SSE. We first characterize the cactus with the minimum SSE among all n-vertex cacti. As an application, we obtain the tree and unicyclic graph with the minimum SSE. Then we give explicit formulas for SSE of double graph and iterated double graph of a given nontrivial connected graph.