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Valency seven symmetric graphs of order 2 p q

Xiao-Hui Hua, Li Chen (2018)

Czechoslovak Mathematical Journal

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, all connected valency seven symmetric graphs of order 2 p q are classified, where p , q are distinct primes. It follows from the classification that there is a unique connected valency seven symmetric graph of order 4 p , and that for odd primes p and q , there is an infinite family of connected valency seven one-regular graphs of order 2 p q with solvable automorphism groups, and there are four sporadic ones...

Valuations of lines

Josef Mlček (1992)

Commentationes Mathematicae Universitatis Carolinae

We enlarge the problem of valuations of triads on so called lines. A line in an e -structure 𝔸 = 〈 A , F , E 〉 (it means that 〈 A , F 〉 is a semigroup and E is an automorphism or an antiautomorphism on 〈 A , F 〉 such that E ∘ E = 𝐈𝐝 ↾ A ) is, generally, a sequence 𝔸 ↾ B , 𝔸 ↾ U c , c ∈ 𝐅𝐙 (where 𝐅𝐙 is the class of finite integers) of substructures of 𝔸 such that B ⊆ U c ⊆ U d holds for each c ≤ d . We denote this line as 𝔸 ( U c , B ) c ∈ 𝐅𝐙 and we say that a mapping H is a valuation of the line 𝔸 ( U c , B ) c ∈ 𝐅𝐙 in a line 𝔸 ^ ( U ^ c , B ^ ) c ∈ 𝐅𝐙 if it is, for each c ∈ 𝐅𝐙 , a valuation of the triad 𝔸 ( U c , B ) in 𝔸 ^ ( U ^ c , B ^ ) . Some theorems on an existence of...

Valued Groups.

Mikko Saarimäki, Pekka Sorjonen (1992)

Mathematica Scandinavica

Vanishing of the first reduced cohomology with values in an L p -representation

Romain Tessera (2009)

Annales de l’institut Fourier

We prove that the first reduced cohomology with values in a mixing L p -representation, 1 < p < ∞ , vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group has no first reduced ℓ p -cohomology. As a byproduct, we prove a conjecture by Pansu. Namely, the first reduced L p -cohomology on homogeneous, closed at infinity, Riemannian manifolds vanishes. We also...

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