Turan Graphs, Stability Number, and Fibonacci Index

📝 Abstract

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur'an graphs frequently appear in extremal graph theory. We show that Tur'an graphs and a connected variant of them are also extremal for these particular problems.

💡 Analysis

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur'an graphs frequently appear in extremal graph theory. We show that Tur'an graphs and a connected variant of them are also extremal for these particular problems.

📄 Content

arXiv:0802.3284v1 [cs.DM] 22 Feb 2008 Tur´an Graphs, Stability Number, and Fibonacci Index V´eronique Bruy`ere∗ Hadrien M´elot∗,† 22 February, 2008 Abstract. The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur´an graphs frequently appear in extremal graph theory. We show that Tur´an graphs and a connected variant of them are also extremal for these particular problems. Keywords: Stable sets; Fibonacci index; Merrifield-Simmons index; Tur´an graph; α-critical graph. 1 Introduction The Fibonacci index F(G) of a graph G was introduced in 1982 by Prodinger and Tichy [20] as the number of stable sets in G. In 1989, Merrifield and Simmons [16] introduced independently this parameter in the chemistry literature1. They showed that there exist correlations between the boiling point and the Fibonacci index of a molecular graph. Since, the Fibonacci index has been widely studied, especially during the last few years. The majority of these recent results appeared in chemical graph theory [12,13,21,23–25] and in extremal graph theory [9,11,17–19]. In this literature, several results are bounds for F(G) among graphs in particular classes. Lower and upper bounds inside the classes of general graphs, connected graphs, and trees are well known (see Section 2). Several authors give a characterization of trees with maximum Fibonacci index inside the class T (n, k) of trees with order n and a fixed parameter k. For example, Li et al. [13] determine such trees when k is the diameter; Heuberger and Wagner [9] when k is the maximum degree; and Wang et al. [25] when k is the number of pending vertices. Unicyclic graphs are also investigated in similar ways [17,18,24]. The Fibonacci index and the stability number of a graph are both related to stable sets. Hence, it is natural to use the stability number as a parameter to determine bounds for F(G). Let G(n, α) and C(n, α) be the classes of – respectively general and connected – graphs with order n and stability number α. The lower bound for the Fibonacci index is known for graphs in these classes. Indeed, Pedersen and Vestergaard [18] give a simple proof to show that if G ∈G(n, α) or G ∈C(n, α), then F(G) ≥2α + n −α. Equality occurs if and only if G is a complete split graph (see Section 2). In this article, we determine upper bounds for F(G) in the classes G(n, α) and C(n, α). In both cases, the bound is tight for every possible value of α and n and the extremal graphs are characterized. A Tur´an graph is the union of disjoint balanced cliques. Tur´an graphs frequently appear in extremal graph theory. For example, the well-known Theorem of Tur´an [22] states that these graphs have minimum size inside G(n, α). We show in Section 3 that Tur´an graphs have also maximum Fibonacci ∗Department of Theoretical Computer Science, Universit´e de Mons-Hainaut, Avenue du Champ de Mars 6, B-7000 Mons, Belgium. †Charg´e de Recherches F.R.S.-FNRS. Corresponding author. E-mail: hadrien.melot@umh.ac.be. 1The Fibonacci index is called the Fibonacci number by Prodinger and Tichy [20]. Merrifield and Simmons introduced it as the σ-index [16], also known as the Merrifield-Simmons index. 1 index inside G(n, α). Observe that removing an edge in a graph strictly increases its Fibonacci index. Indeed, all existing stable sets remain and there is at least one more new stable set: the two vertices incident to the deleted edge. Therefore, we might have the intuition that the upper bound for F(G) is a simple consequence of the Theorem of Tur´an. However, we show that it is not true (see Sections 2 and 5). The proof uses structural properties of α-critical graphs. Graphs in C(n, α) which maximize F(G) are characterized in Section 4. We call them Tur´an- connected graphs since they are a connected variant of Tur´an graphs. It is interesting to note that these graphs again minimize the size inside C(n, α). Hence, our results lead to questions about the relations between the Fibonacci index, the stability number, the size and the order of graphs. These questions are summarized in Section 5. 2 Basic properties In this section, we suppose that the reader is familiar with usual notions of graph theory (we refer to Berge [1] for more details). First, we fix our terminology and notation. We then recall the notion of α-critical graphs and give properties of such graphs, used in the next sections. We end with some basic properties of the Fibonacci index of a graph. 2.1 Notations Let G = (V, E) be a simple and undirected graph order n(G) = |V | and size m(G) = |E|. For a vertex v ∈V (G), we denote by N(v) the neighborhood of v; its closed neighborhood is defined as N(v) = N(v) ∪{v}. The degree of a vertex v is denoted by d(v) and the maximum degree of G by ∆(G). We use notatio

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