The stability index of graphs

• Published in: Combinatorial Mathematics pp. 29 - 52
• Main Author: Grant, Douglas D.
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 27.06.2006
• Series: Lecture Notes in Mathematics
• ISBN: 3540069038; 9783540069034
• ISSN: 0075-8434; 1617-9692
• DOI: 10.1007/BFb0057373

If G is a graph with vertex set V(G) and (vertex) automorphism group γ(G), then a sequence s={vπ(i)}i=1k of distinct vertices of G is a partial stabilising sequence for G if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \left( {G_{S_n } } \right) = \Gamma \left( G \right)_{S_n } $$\end{document}for n = 1,...,k. Here S is the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigcup\limits_{i = 1}^n {V_{\pi (i)} ,G_{S_n } } $$\end{document}is the subgraph of G induced by the subset V(G) — Sn of V(G) and γ(G)Sn is the group of permutations in γ(G) which fix each vertex in Sn, considered as acting on V(G) − Sn. 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.