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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
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