On the eccentricity-based invariants of uniform hypergraphs

Abstract

Let G =(V, E) be a simple connected hypergraph with V the vertex set and E the edge set, respectively. The eccentricity of vertex v refers to the farthest distance of vertex v from other vertices of G, denoted by ?G(v). The eccentric adjacency index (EAI) of G is described as ?ad(G) = ?u?V(G) SG(u)/?G(u), where SG(u) = ? v?NG(u) dG(v). In this work, we consider the gerneralation of the EAI for hypergraphs to draw several conclusions related to extremal problems to EAI. We first propose several bounds on the EAI of k-uniform hypertrees with fixed maximum degree, diameter and edges, respectively, and characterize the corresponding extremal k-uniform hypertrees. Then we investigate the relationsip between EAI and the adjacent eccentric distance sum. Finally, we present the upper bounds for the difference between the eccentricity distance sum and eccentric connectivity index in the k-uniform hypergraph with diameter 2. It generalizes the previous results of the current authors from the simple graphs to hypergraphs for graph parameters based on eccentricity.