Defensive -alliances in graphs

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