On defensive alliances and line graphs

Let ΓΓ be a simple graph of size mm and degree sequence δ1≥δ2≥⋯≥δnδ1≥δ2≥⋯≥δn. Let L(Γ)L(Γ) denote the line graph of ΓΓ. The aim of this work is to study mathematical properties of the alliance number, a(L(Γ))a(L(Γ)), and the global alliance number, γa(L(Γ))γa(L(Γ)), of the line graph of a simple graph. We show that ⌈δn+δn−1−12⌉≤a(L(Γ))≤δ1. In particular, if ΓΓ is a δδ-regular graph (δ>0δ>0), then a(L(Γ))=δa(L(Γ))=δ, and if ΓΓ is a (δ1,δ2)(δ1,δ2)-semiregular bipartite graph, then a(L(Γ))=⌈δ1+δ2−12⌉. As a consequence of the study we compare a(L(Γ))a(L(Γ)) and a(Γ)a(Γ), and we characterize the graphs having a(L(Γ))<4a(L(Γ))<4. Moreover, we show that the global-connected alliance number of L(Γ)L(Γ) is bounded by γca(L(Γ))≥⌈D(Γ)+m−1−1⌉, where D(Γ)D(Γ) denotes the diameter of ΓΓ, and we show that the global alliance number of L(Γ)L(Γ) is bounded by γa(L(Γ))≥⌈2mδ1+δ2+1⌉. The case of strong alliances is studied by analogy.