A construction for infinite families of semisymmetric graphs revealing their full automorphism group

Abstract

We give a general construction leading to different non-isomorphic families of connected q-regular semisymmetric graphs of order 2q (n+1) embedded in , for a prime power q=p (h) , using the linear representation of a particular point set of size q contained in a hyperplane of . We show that, when is a normal rational curve with one point removed, the graphs are isomorphic to the graphs constructed for q=p (h) in Lazebnik and Viglione (J. Graph Theory 41, 249-258, 2002) and to the graphs constructed for q prime in Du et al. (Eur. J. Comb. 24, 897-902, 2003). These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For qa parts per thousand yenn+3 or q=p=n+2, na parts per thousand yen2, we obtain their full automorphism group from our construction by showing that, for an arc , every automorphism of is induced by a collineation of the ambient space . We also give some other examples of semisymmetric graphs for which not every automorphism is induced by a collineation of their ambient space

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.…”

“…As pointed out in [1], the incidence system T * n−1 (K) is dual, in the sense of interchanging the roles of points and lines, to the system (actually several isomorphic systems) described in [3]. Of course, dual systems give rise to isomorphic bipartite graphs, so [1] and [3] are studying the same bipartite graphs.…”

The isomorphism problem for linear representations and their graphs The isomorphism problem for linear representations and their graphs

“…Moreover, consider the linear representation T * 2 (K) where q = 2; we checked by computer that whatever point set K you take in H ∞ = PG(2, 2), the full automorphism group of T * 2 (K) will always be larger than the automorphism group induced by collineations of PG (3,2). For q = 2, an element of L contains only two points of P. If K = H ∞ , clearly a permutation of P will always induce an automorphism of T * n (K).…”

“…One of the reasons to generalise the answer to (Q) for incidence graphs of linear representations is the paper [3], where we use them to construct new mutually non-isomorphic infinite families of semisymmetric graphs, i.e. regular edge-transitive, but not vertextransitive graphs.…”

On Some Cycles in Wenger Graphs

“…A more detailed study, see [12], also showed that W 1 (q) is vertex-transitive for all q; W 2 (q) is vertex-transitive for even q; and for any n ≥ 3 and q ≥ 3, and for n = 2 and odd q, the graphs W n (q) are not vertex-transitive. For a recent generalization of these results, see Cara, Rottey and Voorde [5]. Another result of [12] is that W n (q) is connected when 1 ≤ n ≤ q − 1, and disconnected when n ≥ q, in which case it has q n−q+1 components, each isomorphic to W q−1 (q).…”