On defensive alliances and line graphs

Let $\mathrm{\Gamma }$ be a simple graph of size $m$ and degree sequence $\delta_1\ge \delta_2\ge … \ge \delta_n$\mathrm{.}Let${\cal L}(\Gamma)$denotesthelinegraphof$\Gamma$. The aim of this paper is to study mathematical properties of the alliance number, $a(\mathcal{L}(\mathrm{\Gamma })$, and the global alliance number, ${\gamma }_a(\mathcal{L}(\mathrm{\Gamma }))$, of the line graph of a simple graph. We show that $\lceil\frac{\delta_{n}+\delta_{n-1}-1}{2}\rceil \le {a}({\cal L}(\Gamma))\le \delta_1.$Inparticular,if$\Gamma$isa$\delta$-regular graph ($\delta >0$), then $a(\mathcal{L}(\mathrm{\Gamma }))=\delta$, and if $\mathrm{\Gamma }$ is a $({\delta }_1,{\delta }_2)$-semiregular bipartite graph, then $a({\cal L}(\Gamma))=\lceil \frac{\delta_1+\delta_2-1}{2} \rceil$. As a consequence of the study we compare $a(\mathcal{L}(\mathrm{\Gamma }))$ and $a(\mathrm{\Gamma })$, and we characterize the graphs having $a(\mathcal{L}(\mathrm{\Gamma }))<4$. Moreover, we show that the global-connected alliance number of $\mathcal{L}(\mathrm{\Gamma })$ is bounded by ${\gamma }_{ca}(\mathcal{L}(\mathrm{\Gamma }))\ge \lceil \sqrt{D(\mathrm{\Gamma })+m-1}-1\rceil ,$ where $D(\mathrm{\Gamma })$ denotes the diameter of $\mathrm{\Gamma }$, and we show that the global alliance number of $\mathcal{L}(\mathrm{\Gamma })$ is bounded by $\gamma_{a}({\cal L}(\Gamma))\geq \lceil\frac{2m}{\delta_{1}+\delta_{2}+1}\rceil$. The case of strong alliances is studied by analogy.