The stability index of graphs

Abstract

If G is a graph with vertex set V(G) and (vertex) automorphism group γ(G), then a sequence s={v_π(i)} ^k_i=1 of distinct vertices of G is a partial stabilising sequence for G if $\Gamma \left( {G_{S_n } } \right) = \Gamma \left( G \right)_{S_n }$for n = 1,...,k. Here S is the set $\bigcup\limits_{i = 1}^n {V_{\pi (i)} ,G_{S_n } }$is the subgraph of G induced by the subset V(G) — S_n of V(G) and γ(G)S_n is the group of permutations in γ(G) which fix each vertex in S_n, considered as acting on V(G) − S_n. The stability index of G, s.i. (G), is the maximum cardinality of a partial stabilising sequence for G; thus s.i. (G) = 0 if and only if G is not semi-stable (see [6]) and s.i. (G) = ¦v(G)¦ if and only if G is stable (see [4]).

The stability coefficient of G is s.c. (G) = s.i. (G)/¦V(G)¦. Making use of the above concepts, we characterise unions and joins of graphs which are semi-stable and enumerate trees with given stability index. Finally we investigate the problem of finding graphs with a given rational number as stability coefficient.