Endomorphisms of quadratic forms graph in characteristic two
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Endomorphisms of quadratic forms graph in characteristic two Li-Ping Huang1 Received: 19 April 2018 / Accepted: 11 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Let q be a power of 2, and let Fq be the finite field with q elements. The quadratic forms graph, denoted by Q(n, q) where n ≥ 2, has all quadratic forms on Fqn as vertices and two vertices f and g are adjacent if rk( f − g) = 1 or 2. A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. A graph G is a core if every endomorphism of G is an automorphism. We prove that Q(n, q) is a pseudo-core and Q(2m, q) is a core. Moreover, we gave the smallest eigenvalue of Q(n, q). Keywords Quadratic forms graph · Endomorphism · Pseudo-core · Core · Maximal clique · Eigenvalue Mathematics Subject Classification 05C60 · 05E99 · 05C50
1 Introduction Throughout, let Fq be the finite field with q elements where q is a power of a prime. Write Fq∗ = Fq \{0}. Let Fqn be the vector space whose elements are n-dimensional row vectors over Fq . A quadratic form f on Fqn is a map f : Fqn → Fq such that, for every u, v ∈ Fqn and a, b ∈ Fq , f (au + bv) = a 2 f (u) + b2 f (v) + abB f (u, v) for some bilinear form B f (u, v) =: B f on Fqn . Note that B f is alternating if q is even and symmetric if q is odd. We denote by Rad(B f ) and Rad( f ) the radicals of B f and f , i.e.
Project 11371072 supported by National Natural Science Foundation of China.
B 1
Li-Ping Huang [email protected] School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410004, China
123
Journal of Algebraic Combinatorics
Rad(B f ) = u ∈ Fqn : B f (u, v) = 0, ∀ v ∈ Fqn , Rad( f ) = u ∈ Fqn : f (u) = 0, B f (u, v) = 0, ∀ v ∈ Fqn .
(1.1) (1.2)
The rank of a quadratic form f on Fqn , denoted by rk( f ), is the dimension of the quotient space Fqn /Rad( f ), i.e. rk( f ) = n − dim(Rad( f )). There is a one-to-one correspondence between quadratic forms on Fqn and homogeneous polynomials of degree 2 in n variables over Fq . All graphs are simple [4] and finite. For a graph Γ , we denote by V (Γ ) and diam(Γ ) the vertex set and the diameter of Γ , respectively. Let dΓ (x, y) (d(x, y) for short) be the distance between vertices x and y in Γ . We write x ∼ y if x, y ∈ V (Γ ) are adjacent. Let |X | be the cardinal number of a set X . The quadratic forms graph, denoted by Q(n, q) where n ≥ 2, has all quadratic forms on Fqn as vertices and two vertices f and g are adjacent if rk( f − g) = 1 or 2. We have (cf. [1]) |V (Q(n, q))| = q n(n+1)/2 . When n = 2, Q(n, q) is a complete graph. The automorphism group of Q(n, q) is well known (cf. [1,16,17]). For any f 0 ∈ V (Q(n, q)) and any invertible n ×n matrix P over Fq , maps ϕ( f ) = f − f 0 and φ( f (x)) = f (x P) for all f = f (x) ∈ V (Q(n, q)) are two automorphisms of Q(n, q). Thus, Q(n, q) is vertex-transitive, but Q(n, q) is not distance-transitive when n > 3 (cf. [1, Proposition 9.6.4]. For a graph Γ and x ∈ V (Γ ), let Γi (x)
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