Symmetric Bipartite Graphs of Prime Valency

Symmetric graphs of order 4p of valency prime

A graph is symmetric or arc-transitive if its automorphism group acts transitively on vertices, edges and arcs. Let p, q be odd primes with p, q ≥ 5 and X a q-valent symmetric graph of order 4p. In this paper, we proved that X K4p with 4p-1=q, X K2p,2p-2pK2 with 2p-1=q, the quotient graph of X is isomorphic to Kp,p and p=q, or K2p and 2p-1=q.

Symmetric Graphs of Valency

In this thesis we classify primitive amalgams of degree (5,2) and thereby obtain a description of symmetric graphs of valency 5. There are 22 such amalgams, and together with this list, we find the presentations for the universal completions of such amalgams and an example of a faithful finite completion in each case. Our methods include group theory, the amalgam method and some modular represe...

Symmetric Bipartite Graphs and Graphs with Loops

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.

A prime factor theorem for bipartite graphs

It has long been known that the class of connected nonbipartite graphs (with loops allowed) obeys unique prime factorization over the direct product of graphs. Moreover, it is known that prime factorization is not necessarily unique in the class of connected bipartite graphs. But any prime factorization of a connected bipartite graph has exactly one bipartite factor. It has become folklore in s...

Primitive Half-transitive Graphs of Valency Twice a Prime