Extremal total eccentricity of k-apex trees

In a simple connected graph G , eccentricity of a vertex is one of the first, distance-based invariants. The eccentricity of a vertex v in a connected graph G is the maximum distance of the vertex v to any other vertex u . The total eccentricity of the graph G is the sum of the all vertex eccentricities. A graph G is called an apex tree if it has a vertex x such that G − x is a tree. In this work we have found the graph having extremal total eccentricity of k -apex trees.