Smallest graphs with given automorphism group

THE AUTOMORPHISM GROUP OF FINITE GRAPHS

Let G = (V,E) be a simple graph with exactly n vertices and m edges. The aim of this paper is a new method for investigating nontriviality of the automorphism group of graphs. To do this, we prove that if |E| >=[(n - 1)2/2] then |Aut(G)|>1 and |Aut(G)| is even number.

Vector representable matroids of given rank with given automorphism group

A simple way of as,;ociating a matroid of prescribed rank with a graph is shown. The matroids so corhstrueted are re.presentable over any sufficiently large field. Their ase is demonstrated by the fallowing result: Given an integer k >~ 3 and a hmction G associating a group with each subsel of a set S, the'e is a matroid M(E), representable ever any sufl]ciendy large field, such that E ~S, and ...

Graphs with given Group

Graphs with smallest forgotten index

The forgotten topological index of a molecular graph $G$ is defined as $F(G)=sum_{vin V(G)}d^{3}(v)$, where $d(u)$ denotes the degree of vertex $u$ in $G$. The first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number $gamma=1,2$, the first through<br /...

Subdirectly Reducible Groups and Edge-minimal Graphs with given Automorphism Group

A group is subdirectly reducible if it has two non-trivial normal subgroups with trivial intersection. Such groups may be an easy case in certain inductive arguments. We prove that every solvable finite group can be generated by at most one subdirectly reducible subgroup together with two subgroups of prime-power order and one element. We also prove that every group of prime-power order can be ...