Hypergraph coverings and Ramanujan Hypergraphs

Abstract

In this paper we investigate Ramanujan hypergraphs by using hypergraph coverings. We first show that the spectrum of a $k$-fold covering $\bar{H}$ of a connected hypergraph $H$ contains the spectrum of $H$, and that it is the union of the spectrum of $H$ and the spectrum of an incidence-signed hypergraph with $H$ as underlying hypergraph if $k=2$, which generalizes Bilu-Linial result on graph coverings. We give a lower bound for the second largest eigenvalue of a $d$-regular hypergraph by universal cover, which generalizes Alon-Boppana bound on $d$-regular graphs and Feng-Li bound on $(d,r)$-regular hypergraphs. By using interlacing family of polynomials, we prove that every $(d,r)$-regular hypergraph has a right-sided Ramanujan $2$-covering, and has a left-sided Ramanujan $2$-covering if the roots of the matching polynomial of its incident graph satisfy some condition. By Ramanujan $2$-coverings, we prove the existence of some families of infinite many left-sided or right-sided $(d,r)$-regular Ramanujan hypergraphs under certain conditions on $d$ and $r$.

Figures (2)

• Figure 2.1: Construction of a $2$-covering of a hypergraph, where an edge of $H$ or $H^\phi$ is represented by the point set of a triangle, and only the blue edge in $(B_H,\phi)$ is assigned to the permutation $(12)$ and all other edges are assigned to the identity
• Figure 2.2: Construction of the universal cover of the hypergraph $H$ in Fig. \ref{['FC']} via the universal cover of its incident graph $B_H$, where in the left graph (the infinite tree $\mathbb{T}_{B_H}$), a vertex represents a non-backtracking walk of $B_H$ from the root $v_2$ to the vertex which is the unique path of $\mathbb{T}_{B_H}$ from $v_2$ to the vertex, e.g. the vertex $v_1$ in the $7$th level represents the walk: $v_2e_1v_3e_2v_2e_1v_1$, a unique path of $\mathbb{T}_{B_H}$ from the root to $v_1$ in the $7$th level

Theorems & Definitions (36)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1