In this paper we study defensive alliances in some regular graphs. We determine which subgraphs could a critical defensive alliance of a graph G induce, if G is 6-regular and the cardinality of the alliance is at most 8. In particular, we study the case of circulant graphs, i.e. Cayley graphs on a cyclic group. The critical defensive alliances of a circulant graph of degree at most 6 are comple...

Let Γ = (V,E) be a simple graph of order n and degree sequence δ1 ≥ δ2 ≥ · · · ≥ δn. For a nonempty set X ⊆ V , and a vertex v ∈ V , δX(v) denotes the number of neighbors v has in X. A nonempty set S ⊆ V is a defensive k-alliance in Γ = (V,E) if δS(v) ≥ δS̄(v) + k, ∀v ∈ S. The defensive k-alliance number of Γ, denoted by ak(Γ), is defined as the minimum cardinality of a defensive k-alliance in Γ...

We study a six-species Lotka-Volterra-type system on different two-dimensional lattices when each species has two superior and two inferior partners. The invasion rates from predator sites to a randomly chosen neighboring prey site depend on the predator-prey pair, whereby cyclic symmetries within the two three-species defensive alliances are conserved. Monte Carlo simulations reveal an unexpec...

A nonempty set S ⊆ V is a defensive k-alliance in Γ = (V,E), k ∈ [−Δ,Δ] ∩ Z, if for every v ∈ S, δS(v) ≥ δS̄(v) + k. An defensive kalliance S is called critical if no proper subset of S is an defensive k-alliances in Γ = (V,E). The upper defensive k-alliance number of Γ, denoted by Ak(Γ), is defined as the maximum cardinality of a critical defensive k-alliance in Γ. In this paper we study the ma...

A defensive k-alliance in a graph G = (V, E) is a set of vertices A ⊆ V such that for every vertex v ∈ A, the number of neighbors v has in A is at least k more than the number of neighbors it has in V − A (k is a measure of the strength of alliance). In this paper, we deal with two types of sets associated with defensive k-alliances; maximum defensive k-alliance free and minimum defensive k-all...

Let Γ be a simple graph of size m and degree sequence δ1 ≥ δ2 ≥ · · · ≥ δn. Let L(Γ) denotes the line graph of Γ. The aim of this paper is to study mathematical properties of the alliance number, a(L(Γ), and the global alliance number, γa(L(Γ)), of the line graph of a simple graph. We show that ⌈