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Groups in which the prime graph is a tree

Maria Silvia Lucido (2002)

Bollettino dell'Unione Matematica Italiana

The prime graph Γ ⁢ G of a finite group G is defined as follows: the set of vertices is π ⁢ G , the set of primes dividing the order of G , and two vertices p , q are joined by an edge (we write p ∼ q ) if and only if there exists an element in G of order p ⁢ q . We study the groups G such that the prime graph Γ ⁢ G is a tree, proving that, in this case, π ⁢ G ≤ 8 .

Hamiltonicity of cubic Cayley graphs

Henry Glover, Dragan Marušič (2007)

Journal of the European Mathematical Society

Following a problem posed by Lovász in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a ( 2 , s , 3 ) -presentation, that is, for groups G = 〈 a , b ∣ a 2 = 1 , b s = 1 , ( a b ) 3 = 1 , ⋯ 〉 generated by an involution a and an element b of order s ≥ 3 such that their product a b has order 3 . More precisely, it is shown that the Cayley graph X = Cay ( G , { a , b , b - 1 } ) has a Hamilton cycle when | G | (and thus s ) is congruent to 2 modulo 4, and has a long cycle missing...

Hexavalent ( G , s ) -transitive graphs

Song-Tao Guo, Xiao-Hui Hua, Yan-Tao Li (2013)

Czechoslovak Mathematical Journal

Let X be a finite simple undirected graph with a subgroup G of the full automorphism group Aut ( X ) . Then X is said to be ( G , s ) -transitive for a positive integer s , if G is transitive on s -arcs but not on ( s + 1 ) -arcs, and s -transitive if it is ( Aut ( X ) , s ) -transitive. Let G v be a stabilizer of a vertex v ∈ V ( X ) in G . Up to now, the structures of vertex stabilizers G v of cubic, tetravalent or pentavalent ( G , s ) -transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers G v of connected hexavalent ( G , s ) -transitive...

Integral Cayley Sum Graphs and Groups

Xuanlong Ma, Kaishun Wang (2016)

Discussiones Mathematicae Graph Theory

For any positive integer k, let Ak denote the set of finite abelian groups G such that for any subgroup H of G all Cayley sum graphs CayS(H, S) are integral if |S| = k. A finite abelian group G is called Cayley sum integral if for any subgroup H of G all Cayley sum graphs on H are integral. In this paper, the classes A2 and A3 are classified. As an application, we determine all finite Cayley sum integral groups.

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