Adjacency preservers, symmetric matrices, and cores

Orel, Marko

doi:10.1007/s10801-011-0318-0

Cited by 13 publications

(7 citation statements)

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“…In [30] it was shown that the graph formed by hermitian matrices over a finite field is a core, so its endomorphisms are characterized by the fundamental theorem. Same type of a result was obtained for n × n symmetric matrices over a finite field if n ≥ 3, while in 2 × 2 case the graph possesses a complete core [32]. Recently it was proved that any (finite) distance transitive graph is either a core or it has a complete core [12], though it is mentioned that is often difficult to decide which of the two possibilities occurs.…”

Section: Introductionmentioning

confidence: 68%

Adjacency preservers on invertible hermitian matrices I

Orel

2016

Linear Algebra and its Applications

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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

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Section: Introductionmentioning

confidence: 68%

Adjacency preservers on invertible hermitian matrices I

Orel

2016

Linear Algebra and its Applications

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“…Thus, the fundamental theorem of geometry of F q m×n can be formulated in terms of graph endomorphisms on (F q m×n , ∼). In [16,17], the author characterized the graph endomorphisms on symmetric matrix graphs and hermitian matrix graphs over a finite field. In Section 3, we will characterize graph endomorphisms on matrix graphs using results in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Graphs associated with matrices over finite fields and their endomorphisms

Huang

et al. 2014

Linear Algebra and its Applications

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Abstract. Let F m×n be the set of m × n matrices over a field F. Consider a graph G = (F m×n , ∼)with F m×n as the vertex set such that two vertices A, B ∈ F m×n are adjacent if rank(A − B) = 1. We study graph properties of G when F is a finite field. In particular, G is a regular connected graph with diameter equal to min{m, n}; it is always Hamiltonian. Furthermore, we determine the independence number, chromatic number and clique number of G. These results are used to characterize the graph endomorphisms of G, which extends Hua's fundamental theorem of geometry on F m×n .

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“…The proof was later shortened in the thesis [16] by applying [8,Theorem 4.1]. Analogous result for symmetric matrices over a finite field was obtained in [15]. Here the graph is a core, unless we consider 2 ⇥ 2 symmetric matrices in which case the core of a graph is complete.…”

Section: Introductionmentioning

confidence: 90%