On the eccentricity-based graph 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 εG(v). The eccentric adjacency index (EAI) of G is described as ξ^{ad} (G) = \sum_{u∈V (G)} S_G(u)/ε_G(u) , where S_G(u) = \sum_{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.