Unit-graphs and special unit-digraphs on matrix rings

Yeşi̇m Demiroğlu Karabulut

doi:10.1515/forum-2017-0262

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Unit-graphs and special unit-digraphs on matrix rings

Yeşi̇m Demiroğlu Karabulut

Published/Copyright: June 30, 2018

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From the journal Forum Mathematicum Volume 30 Issue 6

Abstract

We use the unit-graphs and the special unit-digraphs on matrix rings to show that every n×n nonzero matrix over 𝔽q can be written as a sum of two SLn-matrices when n>1. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and we prove that if X is a subset of Mat2⁡(𝔽q) with size |X|>2⁢q3⁢q/(q-1), then X contains at least two distinct matrices whose difference has determinant α for any α∈𝔽q∗. Using this result, we also prove a sum-product type result: if A,B,C,D⊆𝔽q satisfy |A|⁢|B|⁢|C|⁢|D|4=Ω⁢(q0.75) as q→∞, then (A-B)⁢(C-D) equals all of 𝔽q∗. In particular, if A is a subset of 𝔽q with cardinality |A|>32⁢q3/4, then the subset (A-A)⁢(A-A) equals all of 𝔽q. We derive some identities involving character sums of the entries of 2×2 matrices over finite fields. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.

Keywords: Spectral graph theory; matrix rings; sum-product problem

MSC 2010: 05C50; 16U60; 15B33

Communicated by Christopher D. Sogge

A On spectral theory of Cayley digraphs

Let H be a finite abelian group and let S be a subset of H. The Cayley digraphCay⁡(H,S) is the digraph whose vertex set is H and there is an edge from u to v (denoted by u→v) if and only if v-u∈S. By definition, Cay⁡(H,S) is a simple digraph with d+⁢(u)=d-⁢(u)=|S|. Furthermore, if we have a Cayley digraph, then we can find its spectrum easily by using characters in representation theory; see [11] for a rigorous treatment of character theory. A function χ:H→ℂ is a character of H if χ is a group homomorphism of H into the multiplicative group ℂ∗. If χ⁢(h)=1 for every h∈H, we say χ is the trivial character. The following theorem is a very important well-known fact; see, e.g., [2].

Theorem A.1.

Let A be an adjacency matrix of a Cayley digraph Cay⁡(H,S). Let χ be a character on H. Then the vector (χ⁢(h))h∈H is an eigenvector of A with eigenvalue ∑s∈Sχ⁢(s). In particular, the trivial character corresponds to the trivial eigenvector 1 with eigenvalue |S|.

Proof.

Let u1,u2,…,un be an ordering of the vertices of the digraph and let 𝔸 correspond to this ordering. Pick any ui. Then we have

∑j=1n𝔸i⁢j⁢χ⁢(uj)=∑ui→ujχ⁢(uj)=∑s∈Sχ⁢(ui+s)=∑s∈Sχ⁢(ui)⁢χ⁢(s)=[∑s∈Sχ⁢(s)]⁢χ⁢(ui).∎

Notice that we get |H| many eigenvectors as demonstrated in the previous theorem, and they are all distinct since they are orthogonal by character orthogonality; see [11]. This means we know all of the eigenvectors of a Cayley digraph explicitly if we know all of the characters of H. Theorem A.2 (viz. spectral gap theorem) below is a very important and widely used tool in graph theory by itself; see [3] for the proof.

Theorem A.2 (Spectral gap theorem for Cayley digraphs).

Let Cay⁡(H,S) be a Cayley digraph of order n. Let {χi}i=1,2,…,n be the set of all distinct characters on H such that χ1 is the trivial one. Define

n∗=n|S|⁢(max2⩽i⩽n⁡∥∑s∈Sχi⁢(s)∥)

and let X and Y be subsets of vertices of Cay⁡(H,S). If |X|⁢|Y|>n∗, then there exists a directed edge between a vertex in X and a vertex in Y. In particular, if |X|>n∗, then there exist at least two distinct vertices of X with a directed edge between them.

Acknowledgements

I would like to thank my advisers, Professor Jonathan Pakianathan and Professor David Covert for suggesting this problem and for enlightening discussions. I also would like to thank the anonymous referee very much for his/her valuable contributions.

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Received: 2017-12-21

Revised: 2018-05-11

Published Online: 2018-06-30

Published in Print: 2018-11-01

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https://doi.org/10.1515/forum-2017-0262

Keywords for this article

Spectral graph theory; matrix rings; sum-product problem