Primoz Potocnik

Cubic vertex-transitive graphs on up to 1280 vertices

A graph is called cubic and tetravalent if all of its vertices have valency 3 and 4, respectively. It is called vertex-transitive and arc-transitive if its automorphism group acts transitively on its vertex-set and on its arc- set, respectively. In this paper, we combine some new theoretical results with computer calculations to construct all cubic vertex-transitive graphs of order at most 1280. In the process, we also construct all tetravalent arc-transitive graphs of order at most 640.

Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs

The main result of this paper is that, if ? is a connected 4-valent G-arc-transitive graph and v is a vertex of ?, then either ? is part of a well-understood infinite family of graphs, or | G v | ? 2 4 3 6 or 2 | G v | log 2 ? ( | G v | / 2 ) ? | V ? | and that this last bound is tight. As a corollary, we get a similar result for 3-valent vertex-transitive graphs.

Semisymmetry of Generalized Folkman Graphs

A regular edge- but not vertex-transitive graph is said to be semisymmetric. The study of semisymmetric graphs was initiated by Folkman, who, among others, gave constructions of several infinite families such graphs . In this paper a generalization of his construction for orders a multiple of 4 is proposed, giving rise to some new families of semisymmetric graphs. In particular, one associated with the cyclic group of order n, n? 5, which belongs to the class of tetracirculants, that is, graphs admitting an automorphism with precisely four orbits, all of the same length. Semisymmetry properties of tetracirculants are investigated in greater detail, leading to a classification of all semisymmetric graphs of order 4 p, where p is a prime.

TETRAVALENT ARC-TRANSITIVE GRAPHS WITH UNBOUNDED VERTEX-STABILIZERS

It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilisers. However, beside a well known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilisers, each significant for its own reason.

On 2-arc-transitive Cayley graphs of Abelian groups

A 2-arc in a graph X is a sequence of three distinct vertices of graph X where the first two and the last two are adjacent. A graph X is 2-arc-transitive if its automorphism group acts transitively on the set of 2-arcs of X. Some properties of 2-arc-transitive Cayley graphs of Abelian groups are considered. It is also proved that the set of generators of a 2-arc-transitive Cayley graph of an Abelian group which is not a circulant contains no elements of odd order.

Wiener index of iterated line graphs of trees homeomorphic to H

Abstract This is fourth paper out of five in which we completely solve a problem of Dobrynin, Entringer and Gutman. Let G be a graph. Denote by L i ( G ) its i -iterated line graph and denote by W ( G ) its Wiener index. Moreover, denote by H a tree on six vertices, out of which two have degree 3 and four have degree 1. Let j ≥ 3 . In previous papers we proved that for every tree T , which is not homeomorphic to a path, claw K 1 , 3 and H , it holds W ( L j ( T ) ) > W ( T ) . Here we prove that W ( L 4 ( T ) ) > W ( T ) for every tree T homeomorphic to H . As a consequence, we obtain that with the exception of paths and the claw K 1 , 3 , for every tree T it holds W ( L i ( T ) ) > W ( T ) whenever i ≥ 4 .

Asymptotic enumeration of vertex-transitive graphs of fixed valency

Abstract Let G be a group and let S be an inverse-closed and identity-free generating set of G . The Cayley graph Cay ( G , S ) has vertex-set G and two vertices u and v are adjacent if and only if u v − 1 ∈ S . Let C A Y d ( n ) be the number of isomorphism classes of d -valent Cayley graphs of order at most n . We show that log ⁡ ( C A Y d ( n ) ) ∈ Θ ( d ( log ⁡ n ) 2 ) , as n → ∞ . We also obtain some stronger results in the case d = 3 .

On the order of arc-stabilisers in arc-transitive graphs with prescribed local group

Let

Preface: Algorithmic Graph Theory on the Adriatic Coast

\Gamma

The Wiener index in iterated line graphs

be a connected