Groups and polar graphs
Groups generated by k-root subgroups.
Groups in which the prime graph is a tree
The prime graph of a finite group is defined as follows: the set of vertices is , the set of primes dividing the order of , and two vertices , are joined by an edge (we write ) if and only if there exists an element in of order . We study the groups such that the prime graph is a tree, proving that, in this case, .
Groups with the same prime graph as an almost sporadic simple group.
Growth functions and automatic groups.
Growth in
Hamiltonicity of cubic Cayley graphs
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 -presentation, that is, for groups generated by an involution and an element of order such that their product has order . More precisely, it is shown that the Cayley graph has a Hamilton cycle when (and thus ) is congruent to 2 modulo 4, and has a long cycle missing...
Hexavalent -transitive graphs
Let be a finite simple undirected graph with a subgroup of the full automorphism group . Then is said to be -transitive for a positive integer , if is transitive on -arcs but not on -arcs, and -transitive if it is -transitive. Let be a stabilizer of a vertex in . Up to now, the structures of vertex stabilizers of cubic, tetravalent or pentavalent -transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers of connected hexavalent -transitive...
Hicas of length .
Homomorphisms of triangle groups with large injectivity radius.
Homotopy equivalence of isospectral graphs.
Identifying graph automorphisms using determining sets.
Impossible Specifications for the Modular Group.
Infinitely many hypermaps of a given type and genus.
Integral Cayley graphs defined by greatest common divisors.
Integral Cayley graphs over Abelian groups.
Integral Cayley Sum Graphs and Groups
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.
Intersection graphs of finite abelian groups
Intersection graphs of subgroups of finite groups
In this paper we classify finite groups with disconnected intersection graphs of subgroups. This solves a problem posed by Csákány and Pollák.