Abstract:We consider the class of I-graphs Iðn; j; kÞ, which is a generalization over the class of the generalized Petersen graphs. We study different properties of I-graphs, such as connectedness, girth, and whether they are bipartite or vertex-transitive. We give an efficient test for isomorphism of I-graphs and characterize the automorphism groups of I-graphs. Regular bipartite graphs with girth at least 6 can be considered as Levi graphs of some symmetric combinatorial configurations. We consider configurations that arise from bipartite I-graphs. Some of them can be realized in the plane as cyclic astral configurations, i.e., as geometric configurations with maximal isometric symmetry. #

In the paper we show that all combinatorial triangle-free configurations for v ≤ 18 are geometrically realizable. We also show that there is a unique smallest astral (18 3 ) triangle-free configuration and its Levi graph is the generalized Petersen graph G(18, 5). In addition, we present geometric realizations of the unique flag transitive triangle-free configuration (20 3 ) and the unique point transitive triangle-free configuration (21 3 ).

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 action of the stabiliser A v of a given vertex v on the set of arcs incident with v. Specifically, the arc-type is 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 graph with the given partition as its arc-type.

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