The α-normal labeling for generalized directed uniform hypergraphs

Abstract

Let π=(ν1,…,νd) be an ordered partition of the integer m≥1, where ∑i=1dνi=m and νi∈Z+ for all i∈[d]. A π-directed m-uniform hypergraph G consists of a finite vertex set V(G) and a collection of edges E(G). Each edge is an ordered tuple, e=(S1(e),S2(e),…,Sd(e)), of disjoint subsets of vertices such that |Si|=νi, for all i∈[d]. For each edge e, |⋃i=1dSi(e)|=m. Moreover, ⋃e∈E(G)Si(e)=Si(G) and |Si(G)|=ni. Given p=(p1,p2,…,pd)∈(1,∞)d, the p-spectral radius of G is defined asλp(G):=max‖xi‖pi=1,i∈[d]⁡∑e∈E(G)∏i=1d∏v∈Si(e)xi,v, where xi=(xi,1,xi,2,…,xi,ni)∈Rni.In this paper, we develop the α-normal labeling method for calculating λp(G) and some related properties are given. Moreover, we give a new lower bound of the p-spectral radius of π-directed m-uniform hypergraphs for the case ∑i=1dνipi>1 by using the inverse Hölder's inequality.