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- W267300392 abstract "Let X be a finite vertex-transitive graph of valency d , and let A be the full automorphism group of X . Then the arc-type of X is defined in terms of the sizes of the orbits of the stabiliser A v of a given vertex v on the set of arcs incident with v . Such an orbit is said to be self-paired if it is contained in an orbit Δ of A on the set of all arcs of X such that Δ is closed under arc-reversal. The arc-type of X is then the partition of d as the sum n 1 + n 2 + … + n t + ( m 1 + m 1 ) + ( m 2 + m 2 ) + … + ( m s + m s ) , where n 1 , n 2 , …, n t are the sizes of the self-paired orbits, and m 1 , m 1 , m 2 , m 2 , …, m s , m s are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two ‘relatively prime’ graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of 1 + 1 and (1 + 1) , every partition as defined above is realisable , in the sense that there exists at least one vertex-transitive graph with the given partition as its arc-type." @default.
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- W267300392 date "2017-02-12" @default.
- W267300392 modified "2023-09-28" @default.
- W267300392 title "Vertex-transitive graphs and their arc-types" @default.
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