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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 $p \sim q$ ) 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, $|π (G)| \leq 8$ .

Groups with the same prime graph as an almost sporadic simple group.

Khosravi, Behrooz (2009)

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

Growth functions and automatic groups.

Epstein, David B.A., Iano-Fletcher, Anthony R., Zwick, Uri (1996)

Experimental Mathematics

Growth in ${SL}_{3} (ℤ / p ℤ)$

H. A. Helfgott (2011)

Journal of the European Mathematical Society

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 $G = 〈 a, b ∣ a^{2} = 1, b^{s} = 1, {(a b)}^{3} = 1, \dots 〉$ generated by an involution $a$ and an element $b$ of order $s \geq 3$ 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 $v \in V (X)$ 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...

Hicas of length $\leq 4$ .

Miemietz, Vanessa, Turner, Will (2010)

Documenta Mathematica

Homomorphisms of triangle groups with large injectivity radius.

Abas, M. (2003)

Acta Mathematica Universitatis Comenianae. New Series

Homotopy equivalence of isospectral graphs.

Bisson, Terrence, Tsemo, Aristide (2011)

The New York Journal of Mathematics [electronic only]

Identifying graph automorphisms using determining sets.

Boutin, Debra L. (2006)

The Electronic Journal of Combinatorics [electronic only]

Impossible Specifications for the Modular Group.

W. Wilson Stothers (1974)

Manuscripta mathematica

Infinitely many hypermaps of a given type and genus.

Jones, Gareth A., Pinto, Daniel (2010)

The Electronic Journal of Combinatorics [electronic only]

Integral Cayley graphs defined by greatest common divisors.

Klotz, Walter, Sander, Torsten (2011)

The Electronic Journal of Combinatorics [electronic only]

Integral Cayley graphs over Abelian groups.

Klotz, Walter, Sander, Torsten (2010)

The Electronic Journal of Combinatorics [electronic only]

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.

Integral quartic Cayley graphs on Abelian groups.

Abdollahi, A., Vatandoost, E. (2011)

The Electronic Journal of Combinatorics [electronic only]

Intersection graphs of finite abelian groups

Bohdan Zelinka (1975)

Czechoslovak Mathematical Journal

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