On the spectrum of Wenger graphs

Abstract

Let q = p e , where p is a prime and e ≥ 1 is an integer. For m ≥ 1, let P and L be two copies of the (m + 1)-dimensional vector spaces over the finite field F q . Consider the bipartite graph W m (q) with partite sets P and L defined as follows: a point (p) = (p 1 , p 2 , . . . , p m+1 ) ∈ P is adjacent to a line [l] = [l 1 , l 2 , . . . , l m+1 ] ∈ L if and only if the following m equalities hold: l i+1 + p i+1 = l i p 1 for i = 1, . . . , m. We call the graphs W m (q) Wenger graphs. In this paper, we determine all distinct eigenvalues of the adjacency matrix of W m (q) and their multiplicities. We also survey results on Wenger graphs.

Linearized Wenger graphs

“…Thus this derives that the eigenvalues of G are 5) where N Fw = |{u ∈ F q : F w (u) = 0}|. For example, when w = (0, .…”

“…All graph theory notions can be found in Bollobás [2]. Recently, a class of bipartite graphs called Wenger graphs which are defined over F q has attracted a lot of attention because of their nice graphical properties [5,11,12,16,18,19,20,21]. For example, the number of edges of these graphs meets the lower bound of Turán number of the cycle with length 4, 6, 10 [21].…”

“…If g k (x, y) = x k−1 y, k = 2, · · · , m + 1, then the graph is just the original Wenger graph in [5], also denoted by W m (q). It was shown in [11] that the automorphism group of W m (q) acts transitively on each of P and L, and on the set of edges of W m (q).…”

“…In [20], Viglione proved that the diameter of W m (q) is 2m + 2 when 1 ≤ m ≤ q − 1. In [5], Cioabȃ, Lazebnik and Li determined the spectrum of W m (q).…”

Linear representations of finite geometries and associated LDPC codes

“…We will also define words in D analogous to plane words called capacitor words and show that they are codewords that span D. In §4 we apply Theorem 1.1 in particular cases of n and K. In the case where K is a rational normal curve minus its point at infinity the sets P and L form the bipartition of the vertex set of a bipartite graph called the Wenger graph. The Wenger graphs have been studied extensively; their automorphism groups have been found in [1] and their spectra determined in [3]. Our theorem shows that the rank of the adjacency matrix of a Wenger graph over any field F in which q = 0, is the same as the real rank, hence equal to the matrix size minus the multiplicity of zero as an eigenvalue of the adjacency matrix.…”