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Edge-Transitive Lexicographic and Cartesian Products

Wilfried Imrich, Ali Iranmanesh, Sandi Klavžar, Abolghasem Soltani (2016)

Discussiones Mathematicae Graph Theory

In this note connected, edge-transitive lexicographic and Cartesian products are characterized. For the lexicographic product G ◦ H of a connected graph G that is not complete by a graph H, we show that it is edge-transitive if and only if G is edge-transitive and H is edgeless. If the first factor of G ∘ H is non-trivial and complete, then G ∘ H is edge-transitive if and only if H is the lexicographic product of a complete graph by an edgeless graph. This fixes an error of Li, Wang, Xu, and Zhao...

Edge-Transitivity of Cayley Graphs Generated by Transpositions

Ashwin Ganesan (2016)

Discussiones Mathematicae Graph Theory

Let S be a set of transpositions generating the symmetric group Sn (n ≥ 5). The transposition graph of S is defined to be the graph with vertex set {1, . . . , n}, and with vertices i and j being adjacent in T(S) whenever (i, j) ∈ S. In the present note, it is proved that two transposition graphs are isomorphic if and only if the corresponding two Cayley graphs are isomorphic. It is also proved that the transposition graph T(S) is edge-transitive if and only if the Cayley graph Cay(Sn, S) is edge-transitive....

End-faithful spanning trees of countable graphs with prescribed sets of rays

Norbert Polat (2001)

Czechoslovak Mathematical Journal

We prove that a countable connected graph has an end-faithful spanning tree that contains a prescribed set of rays whenever this set is countable, and we show that this solution is, in a certain sense, the best possible. This improves a result of Hahn and Širáň Theorem 1.

Endomorphisms of symbolic algebraic varieties

Misha Gromov (1999)

Journal of the European Mathematical Society

The theorem of Ax says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and investigated in the present paper in the category of proregular mappings of proalgebraic spaces. We show that such maps are surjunctive if they commute with sufficiently large automorphism groups. Of particular interest is the case of proalgebraic varieties over infinite graphs. The paper intends to bring out relations between model theory,...

Expansion and random walks in SL d ( ℤ / p n ℤ ) : I

Jean Bourgain, Alex Gamburd (2008)

Journal of the European Mathematical Society

We prove that the Cayley graphs of SL d ( ℤ / p n ℤ ) are expanders with respect to the projection of any fixed elements in SL d ( ℤ ) generating a Zariski dense subgroup.

Expansion in finite simple groups of Lie type

Emmanuel Breuillard, Ben J. Green, Robert Guralnick, Terence Tao (2015)

Journal of the European Mathematical Society

We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].

Expansion in S L d ( 𝒪 K / I ) , I square-free

Péter P. Varjú (2012)

Journal of the European Mathematical Society

Let S be a fixed symmetric finite subset of S L d ( 𝒪 K ) that generates a Zariski dense subgroup of S L d ( 𝒪 K ) when we consider it as an algebraic group over m a t h b b Q by restriction of scalars. We prove that the Cayley graphs of S L d ( 𝒪 K / I ) with respect to the projections of S is an expander family if I ranges over square-free ideals of 𝒪 K if d = 2 and K is an arbitrary numberfield, or if d = 3 and K = ℚ .

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