On defensive alliances and line graphs

Abstract

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