In this thesis we shall deal with oriented graphs. The motivation for us is a result by Sali and Simonyi (see also a short proof by Gyárfás) where the existence of the decomposition of transitive tournaments on two isomorphic graphs is shown. In this thesis we start to study a problem of packing in transitive tournaments and we consider decompositions and partitions of transitive tournaments. T...

We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors given graph. find semiexplicit Markovian equilibrium for any transitive graph, terms the empirical eigenvalue distribution graph’s normalized Laplacian matrix. This facilitates large-population asymptotics various graph sequences, several sparse and dense ex...

Abstract We prove that, if $$\varGamma $$ Γ is a finite connected 3-valent vertex-transitive, or 4-valent vertex- and edge-transitive graph, then either part of well-understood family graphs, every non-identity automorphism fixes at most 1/3 the edges. This answers question proposed by Primož Potočnik third a...

A graph G is k-ordered if for any sequence of k distinct vertices v1, v2, . . . , vk of G there exists a cycle in G containing these k vertices in the specified order. In 1997, Ng and Schultz posed the question of the existence of 4-ordered 3-regular graphs other than the complete graph K4 and the complete bipartite graph K3,3. In 2008, Meszaros solved the question by proving that the Petersen ...

We study (G, 2)-arc-transitive graphs for innately transitive permutation groups G such that G can be embedded into a wreath product SymΓwr Sl acting in product action on Γ. We find two such connected graphs: the first is Sylvester’s double six graph with 36 vertices, while the second is a graph with 120 vertices whose automorphism group is Aut Sp(4, 4). We prove that under certain conditions n...

The present paper investigates arc-transtive graphs in term of their stability, and shows, somewhat contrary to expectations, that the property of instability is not as rare as previously thought. Until quite recently, the only known example of a finite, arc-transitive vertex-determining unstable graph was the underlying graph of the dodecahedron. This paper illustrates some methods for constru...

A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits of equal size. In this paper, we give a characterization of cubic nonCayley vertex-transitive bi-Cayley graphs over a regular p-group, where p > 5 is a prime. As an application, a classification of cubic non-Cayley vertex-transitive graphs of order 2p3 is given for each prime p.

Any graph which is not vertex transitive has a proper induced subgraph which is unique due to its structure or the way of its connection to the rest of the graph. We have called such subgraph as an anchor. Using an anchor which, in fact, is representative of a graph asymmetry, the reconstruction of that graph reduces to a smaller form of the reconstruction. Therefore, to show that a graph is re...

We survey results on transitive-closure spanners and their applications. Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V,EH) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were studied implicitly in different areas of computer science, and properties of these spanners h...

Let r be a finite connecied regular bipartite 2-arc transitive graph. It is shown that r is a cover of a possibly smaller graph :E, which is also connecied and regular of the same valency as r, and there is a subgroup G of Aut :E such that G is 2-arc transitive on :E and every nontrivial normal subgroup of G has at most two orbits on vertices. Such graphs :E for which the subgroup G has an abel...