By Jack Koolen, Jin Ho Kwak, Ming-Yao Xu

**Applications of team concept to Combinatorics** includes eleven survey papers from foreign specialists in combinatorics, team thought and combinatorial topology. The contributions disguise themes from relatively a various spectrum, resembling layout conception, Belyi features, workforce idea, transitive graphs, standard maps, and Hurwitz difficulties, and current the cutting-edge in those components. **Applications of team conception to Combinatorics** might be important within the learn of graphs, maps and polytopes having maximal symmetry, and is geared toward researchers within the components of staff thought and combinatorics, graduate scholars in arithmetic, and different experts who use team concept and combinatorics.

**Jack Koolen** teaches on the division of arithmetic at Pohang collage of technology and expertise, Korea. His major learn pursuits comprise the interplay of geometry, linear algebra and combinatorics, on which he released 60 papers.

**Jin Ho Kwak** is Professor on the division of arithmetic at Pohang college of technology and expertise, Korea, the place he's director of the Combinatorial and Computational arithmetic middle (Com2MaC). He works on combinatorial topology, ordinarily on overlaying enumeration regarding Hurwitz difficulties and normal maps on surfaces, and released greater than a hundred papers in those areas.

**Ming-Yao Xu** is Professor in division of arithmetic at Peking collage, China. the point of interest in his learn is in finite staff idea and algebraic graph concept. Ming-Yao Xu released over eighty papers on those topics.

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19(1993), 361–407. H. D. Thesis, The University of Western Australia, 1966. H. Li, On isomorphisms of connected Cayley graphs III, Bull. Austral. Math. , 58(1998), 137–145. H. P. Lu and H. Zhang, Tetravalent edge-transitive Cayley graphs with odd number of vertices, J. Combin. Theory B, 96(2006), 164–181. P. Lu, On the automorphism groups of biCayley graphs, Beijing Daxue Xuebao, 39(2003), no. 1, 1–5. P. Q. , 80(2006), 177–187. P. Y. Xu, On the normality of Cayley graphs of order pq, Australasian Journal of Combinatorics, 27 (2003), 81–93.

See [36, 37] for more basic facts and applications of bi-Cayley graphs. 21 Now let X = Cay(G, S) be an edge-transitive directed Cayley graph with G = SS −1 . Then is connected and Aut(X ) acts transitively on its edge set. It follows from [27] that | Aut(X )1 | is bounded by 3 · 27 . 3]. 12 [11]. Let G be a nonabelian simple group and let X = Cay(G, S) be an edgetransitive cubic directed Cayley graph with G = SS −1 . If R(G) is not normal in Aut(X ), then | G | ≤ (3 · 27 )! and G is one of the following groups: An , n = 5, 6, 7, 8, 9, 11, 15, 23, 31, 47, 63, 95, 127, 191, 383; and M11 , M12 , M22 , M23 , J1 , J2 ; and PSL2 (7), PSL2 (11), PSL2 (13), PSU3 (3), PSU4 (3), PSp4 (3); and PSL2 (23 ), PSL2 (24 ), PSL2 (25 ), PSL2 (26 ); and PSL3 (22 ), PSL3 (23 ), PSL3 (24 ), PSL4 (4); and PSL5 (2), PSL5 (2), PSL6 (2), PSL7 (2); and 2 3 PSU3 (4), PSU4 (4), PSU5 (2), PSp4 (4), Sp6 (2), P − 8 (2) and B2 (2 ).

Moreover, if is G-arc-transitive, then G{v,w} contains an element interchanging v and w, and hence so does H . Thus H is arc-transitive on [P] and so ( , P ) is a G-arc-symmetrical factorisation. The following example shows that edge-symmetrical factorisations exist with G = HGv and G either vertex-transitive or vertex-intransitive, and also with HGv = HGw an index two subgroup of G. Note that for arc-symmetrical factorisations only the case G = HGv and G-vertex-transitive occurs as is both G- and H -vertex-transitive in this case.