We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges its two parts. To prove this, we study the relationship between symmetric bipartite graphs and graphs with loops.

We classify the connected-homogeneous digraphs with more than one end. We further show that if their underlying undirected graph is not connected-homogeneous, they are highly-arc-transitive.

A connguration is weakly ag-transitive if its group of automor-phisms acts intransitively on ags but the group of all automorphisms and anti-automorphisms acts transitively on ags. It is shown that weakly ag-transitive conngurations are in one-to-one correspondence with bipartite 1 2-arc-transitive graphs of girth not less than 6. Several innnite families of weakly ag-transitive conngurations a...

A family of arc-transitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of isomorphism between graphs in the family is solved, some combinatorial properties are explored, a...

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs admitting an automorphism groupG that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph B corresponding toB. If in additionG is ...

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...

Let FG(P ) be a functional defined on the set of all the probability distributions on the vertex set of a graph G. We say that G is symmetric with respect to FG(P ) if the uniform distribution on V (G) maximizes FG(P ). Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic numb...

Let Γ be a finite X-symmetric graph with a nontrivial Xinvariant partition B on V (Γ) such that ΓB is a connected (X, 2)-arctransitive graph and Γ is not a multicover of ΓB. A characterization of (Γ,X,B) was given in [20] for the case where |Γ(C) ∩ B| = 2 for B ∈ B and C ∈ ΓB(B). This motivates us to investigate the case where |Γ(C) ∩ B| = 3, that is, Γ[B,C] is isomorphic to one of 3K2, K3,3 − ...