Adjacency preservers on invertible hermitian matrices I

Abstract

Hua's fundamental theorem of geometry of hermitian matrices characterizes all bijective maps on the space of all hermitian matrices, which preserve adjacency in both directions. In this and subsequent paper we characterize maps on the set of all invertible hermitian matrices over a finite field, which preserve adjacency in one direction. This and author's previous result are used to obtain two new results related to maps that preserve the `speed of light' on finite Minkowski space-time. In this first paper it is shown that maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are necessarily bijective, so the corresponding graph on invertible hermitian matrices, where edges are defined by the adjacency relation, is a core. Besides matrix theory, the proof relies on results from several other mathematical areas, including spectral and chromatic graph theory, and projective geometry

On Generalizations of the Petersen Graph and the Coxeter Graph

“…Here the graph is a core, unless we consider 2 ⇥ 2 symmetric matrices in which case the core of a graph is complete. Very recently, the present author showed that the graph, which is formed by invertible hermitian matrices over a field F q 2 with q 2 elements, is a core, provided that q > 4 [13]. In this paper we prove the same result for q = 2.…”

“…Then y j := Q † j x j satisfies y † j y j = 0. By Lemma 13 in case of symmetric matrices, and by [13,Lemma 2.3] applied at 1⇥n matrices y ⇤ 1 , y ⇤ 2 in hermitian case, there is an invertible matrix P such that P † = P 1 and P y 1 = y 2 . If R := (Q 1 2 ) † P Q † 1 , then the automorphism (X) = RXR † satisfies (A 1 ) = A 2 and (B 1 ) = B 2 , so the graph is arc-transitive.…”

“…In this paper we prove the same result for q = 2. The proof demands di↵erent techniques from those in [13], since the graph is without triangles when q = 2. In contrast, when q > 3, any two adjacent vertices determine a maximal clique on either q or q 1 vertices.…”

Adjacency preservers on invertible hermitian matrices II

“…Similarly, we use HGL n (F q 2 ) also to denote the subgraph in Hn(Fq 2 ), which is induced by the set HGL n (F q 2 ). For q ≥ 4 the next lemma is proved in [15] as a corollary of a more Figure 1. A flower of a rank-one matrix in HGL 2 (F 4 2 ).…”

“…An example of a such map is ψ(r) := − g(r), 1, 0, g(r) ⊤ , where g : K 4 → K is an injection. Recall that for Theorem 4.10, the non-existence of such maps in finite field case was essentially proven in Lemma 3.8 of the first paper [15], where spectral graph theory and chromatic number of a graph were used.…”