Let $\Gamma$ be a simple graph of size $m$ and degree sequence $\delta_1\ge \delta_2\ge ... \ge \delta_n$. Let ${ \cal L}(\Gamma)$ denotes the line graph of $\Gamma$. The aim of this paper is to study mathematical properties of the alliance number, ${ a}({ \cal L}(\Gamma)$, and the global alliance number, $\gamma_{ a}({ \cal L}(\Gamma))$, of the line graph of a simple graph. We show that $\lceil\frac{ \delta_{ n}+\delta_{ n-1}-1}{ 2}\rceil \le { a}({ \cal L}(\Gamma))\le \delta_1.$ In particular, if $\Gamma$ is a $\delta$-regular graph ($\delta>0$), then $a({ \cal L}(\Gamma))=\delta$, and if $\Gamma$ 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 $a({ \cal L}(\Gamma))$ and ${ a}(\Gamma)$, and we characterize the graphs having $a({ \cal L}(\Gamma))<4$. Moreover, we show that the global-connected alliance number of ${ \cal L}(\Gamma)$ is bounded by $\gamma_{ ca}({ \cal L}(\Gamma)) \ge \lceil\sqrt{ D(\Gamma)+m-1}-1\rceil,$ where $D(\Gamma)$ denotes the diameter of $\Gamma$, and we show that the global alliance number of ${ \cal L}(\Gamma)$ 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.
Categories: math.CO