On the eccentric connectivity index of uniform hypergraphs

The eccentric connectivity index of a connected hypergraph G with vertex set V(G) is defined as ξc(G)=∑u∈V(G)duηu, where du denotes the degree of u and ηu denotes the eccentricity of u in G. We propose some hypergraph transformations that increase or decrease the eccentric connectivity index of a uniform hypergraph. We determine the unique k-uniform hypertrees with the first two largest eccentric connectivity indices, as well as the unique k-uniform hypertrees with the first three smallest eccentric connectivity indices among k-uniform hypertrees with fixed number of edges. We determine the unique hypertrees with the largest and the smallest eccentric connectivity indices respectively among k-uniform hypertrees with fixed number of edges and fixed diameter. We determine the unique hypertrees with the largest eccentric connectivity index among k-uniform hypertrees with fixed number of edges and fixed maximum degree. We also determine the unique hypergraphs with the largest and the smallest eccentric connectivity indices respectively among k-uniform unicyclic hypergraphs with fixed number of edges.