Defensive k-alliances in graphs

Let G=(V,E)G=(V,E) be a simple graph of order nn and degree sequence δ1≥δ2≥⋯≥δnδ1≥δ2≥⋯≥δn. For a nonempty set X⊆VX⊆V, and a vertex v∈Vv∈V, δX(v)δX(v) denotes the number of neighbors that vv has in XX. A nonempty set S⊆VS⊆V is a defensive   kk-alliance in G=(V,E)G=(V,E) if δS(v)≥δS(v)+k,∀v∈S. The defensive kk-alliance number of GG, denoted by ak(G)ak(G), is defined as the minimum cardinality of a defensive kk-alliance in GG. We study the mathematical properties of ak(G)ak(G). We show that ⌈δn+k+22⌉≤ak(G)≤n−⌊δn−k2⌋ and ak(G)≥⌈n(μ+k+1)n+μ⌉, where μμ is the algebraic connectivity of GG and k∈{−δn,…,δ1}k∈{−δn,…,δ1}. Moreover, we show that for every k,r∈Zk,r∈Z such that −δn≤k≤δ1−δn≤k≤δ1 and 0≤r≤k+δn2, ak−2r(G)+r≤ak(G)ak−2r(G)+r≤ak(G) and, as a consequence, we show that for every k∈{−δn,…,0}k∈{−δn,…,0}, ak(G)≤⌈n+k+12⌉. In the case of the line graph L(G)L(G) of a simple graph GG, we obtain bounds on ak(L(G))ak(L(G)) and, as a consequence of the study, we show that for any δδ-regular graph, δ>0δ>0, and for every k∈{2(1−δ),…,0}k∈{2(1−δ),…,0}, ak(L(G))=δ+⌈k2⌉. Moreover, for any (δ1,δ2δ1,δ2)-semiregular bipartite graph GG, δ1>δ2δ1>δ2, and for every k∈{2−δ1−δ2,…,δ1−δ2}k∈{2−δ1−δ2,…,δ1−δ2}, ak(L(G))=⌈δ1+δ2+k2⌉.