Spectrum of Superhypergraphs via Flows

Abstract

For any $n\in \mathbb{N}$ and given nonempty subset $V$ , the concept of $n$ -superhypergraphs is introduced by Florentin Smarandache based on $P^n\left(V\right)$ ( $n$ -th power set of $V$ ). In this paper, we present the novel concepts supervertices, superedges, and superhypergraph via the concept of flow. This study computes the number of superedges of any given superhypergraphs, and based on the numbers of superedges and partitions of an underlying set of superhypergraph, we obtain the number of all superhypergraphs on any nonempty set. As a main result of the research, this paper is introducing the incidence matrix of superhypergraph and computing the characteristic polynomial for the incidence matrix of superhypergraph, so we obtain the spectrum of superhypergraphs. The flow of superedges plays the main role in computing of spectrum of superhypergraphs, so we compute the spectrum of superhypergraphs in some types such as regular flow, regular reversed flow, and regular two-sided flow. The new conception of superhypergraph and computation of the spectrum of superhypergraphs are introduced firstly in this paper.