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A classification of semisymmetric graphs of order $$2pq$$. (English) Zbl 0944.05051
Let $$X$$ be a bipartite graph with bipartition $$V(X)=U(X)\cup W(X)$$. Let $$A=$$Aut$$(X)$$, $$A^+=\{g\in A\mid U(X)^g=U(X)$$ and $$W(X)^g=W(X)\}$$. Suppose $$G$$ is a subgroup of $$A^+$$. Then $$X$$ is said to be $$G$$-semitransitive, if $$G$$ acts transitively on both $$U(X)$$ and $$W(X)$$, and semitransitive, if $$X$$ is $$A^+$$-semitransitive. We call a graph semisymmetric if it is regular and edge-transitive but not vertex-transitive. This paper gives a classification of semisymmetric graphs of order $$2pq$$ where $$p$$ and $$q$$ are distinct primes. Given a group $$G$$ and a subgroup $$H$$ of $$G$$, we use $$[G:H]$$ to denote the set of right cosets of $$H$$ in $$G$$. Let $$G$$ be a group, let $$L$$ and $$R$$ be subgroups of $$G$$ and let $$D=\bigcup_i Rd_iL$$ be a union of double cosets of $$R$$ and $$L$$ in $$G$$. Define a bipartite graph $$X=B(G,L,R;D)$$ with bipartition $$V(X)=[G:L]\cup [G:R]$$ and edge set $$E(X)=\{(Lg,Rdg)\mid g\in G$$ and $$d\in D\}$$. This graph is called the bi-coset graph of $$G$$ with respect to $$L$$, $$R$$ and $$D$$. Given a $$G$$-semitransitive and $$G$$-edge transitive graph $$X$$ with bipartition $$V(X)=U(X)\cup W(X)$$, where $$|U(X)|=p$$ and $$|W(X)|=pq$$ for two distinct primes $$p$$ and $$q$$, we define a bipartite graph $$\overline X$$ with bipartition $$V(\overline X)=U(\overline X)\cup W(\overline X)$$ as follows: $$U(\overline X)=\{u_i\mid u\in U(X)$$ and $$1\leq i\leq q\}$$, $$W(\overline X)=W(X)$$, $$E(\overline X)=\{(u_i,w)\mid (u,w)\in E(X)$$ and $$1\leq i\leq q\}$$. Suppose $$Y$$ is a semisymmetric graph of order $$2pq$$, where $$p$$ and $$q$$ are distinct primes. Then either $$Y$$ is biprimitive and $$Y$$ is one of the bi-coset graphs $$B(G,L,R;D)$$ with $$G=M_{23}$$ or $$\text{PSL}(2,q)$$, $$q=11,13,23,59,61$$, or $$Y$$ is not biprimitive and $$Y$$ is isomorphic to the graph $$\overline X$$ derived from one of the bi-coset graphs $$B(G,L,R;D)$$ with $$G= \text{PSL}(2,11), \text{PSL}(5,2),S_p$$ or $$G$$ is a semidirect product of $$Z_p$$ by a $$Z_{qk}$$, where $$qk$$ divides $$p-1$$ (Theorem 2.8).

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
##### Keywords:
semisymmetric graphs; permutation groups
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