On the automorphism group of doubled Grassmann graphs
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On the automorphism group of doubled Grassmann graphs MEYSAM ZIAEE Department of Mathematical, Faculty of Basic Sciences, Lorestan University, Khoram-Abad, Iran E-mail: [email protected]
MS received 8 September 2019; revised 20 April 2020; accepted 27 April 2020 Abstract. In this paper, we study a class of regular graphs, which is related to the Grassmann graph. This class of graphs is called the doubled Grassman graph. The Grassmann graph is the class of graphs, which is defined similar to the Johnson graph. This time, however, we are concerned with the subspaces of a vector space, rather than with the subsets of a set. The doubled Grassmann graph is constructed from the Grassmann graph. This graph, was discovered by Biggs and Gardiner. In this paper, we determine the full automorphism group of the doubled Grassmann graph. For determining the automorphism group of the doubled Grassmann graph, we will use the methods used by Mirafzal (Proc. Indian Acad. Sci. (Math Sci.) 129(3) (2019), Art. 34, 8). Keywords.
Automorphism group; Grassmann graph; doubled Grassmann graph.
Mathematics Subject Classification.
05E18, 05C25, 05C69.
1. Introduction and preliminaries In this paper, a graph = (V, E) is considered as a finite undirected simple graph, where V = V () is the vertex-set and E = E() is the edge-set of . The degree of each vertex v ∈ V (), is the number of neighbours of v in and denoted by deg(v). A graph = (V, E) is called k-regular (or regular graph with degree k), if deg(v) = k for every v ∈ V (). Let u, v be two vertices in (connected) graph . Then the length of the shortest path from u to v is called the distance between u, v and denoted by d (u, v) (for clarity, we use d(u, v) instead of d (u, v)). For all the terminologies and notations not defined here, we follow [2–4,8,10]. Let q = p n , where p is a prime. Then we denote the finite field with q elements, by Fq . Also in this paper, the n-dimensional vector space over Fq is denoted by Vn (q). Let k ≤ n, and let Vk be the set of all k-dimensional subspaces (also called k-subspace) of Vn (q). Then the Grassmann graph G(q, n, k) is the graph with vertex-set Vk and two vertices u, w are adjacent if and if dim(u ∩ w) =k − 1. By Brouwer et al. [3], the Grassmann only graph G(q, n, k) has nk q vertices, where nk q is q-array Gaussian binomial coefficient, given by
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(q n − 1) · · · (q n−k+1 − 1) n = . k q (q k − 1) · · · (q − 1) Also, the Grassmann graph G(q, n, k) is the r -regular graph, where r =q
n−k 1
q
k k−1
. q
Let [n] = {1, . . . , n} be a set of size n. Let m be an integer such that 2m ≤ n. Then the Johnson graph J (n, m) is the graph with vertex-set consisting of all m-subsets (subsets of size m) of [n], and two vertices u, v are if and only if |u ∩ v| = m − 1. The adjacent number of vertices of J (n, m) is equal to mn . Furthermore, the Johnson graph J (n, m) is a regular graph with de
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