Edge orbits and edge-deleted subgraphs

Abstract

Abstract. We prove that for any graph G the edge-deleted subgraphs of G partition into two isomorphism classes if and only if the edges of G partition into two orbits under the automorphism group of G.

Uniform Edge Betweenness Centrality

“…We note that a circulant graph has rotational symmetry about its vertices and is therefore vertex-transitive. We will show later that C 15 (1,6) is not edge-transitive, but has uniform edge betweenness centrality and this example can be extended to an infinite class.…”

“…That is, a graph is vertex-transitive (edge-transitive) if its vertices (edges) cannot be distinguished from each other. We will make use of the following alternative definition [1].…”

“…We again used the databases from Brendan McKay [15] and the Wolfram Mathematica 11 code for betweenness testing and found that C 15 (1,6) is the smallest vertex-transitive graph that has uniform edge betweenness centrality but is not edge-transitive. For the sake of completeness the details are given below and in the appendix of this paper.…”

“…(See the appendix for all 51 vertex-transitive graphs on 14 vertices.) Proposition 2.5 C 15 (1,6) is the only graph on 15 vertices which is edge-betweenness-uniform and vertextransitive and not edge-transitive. More specifically, C 15 (1,6) is the smallest graph satisfying these three conditions.…”

“…There are 44 vertex-transitive graphs on 15 vertices, 10 of which are both edge-transitive and vertex-transitive and 33 of which are not edge-betweenness-uniform. The remaining graph is C 15 (1,6), which is edge-betweenness-uniform and vertex-transitive, but not edge-transitive. (See the appendix for all 44 vertextransitive graphs on 15 vertices.…”