Offensive alliances in cubic graphs

An offensive alliance in a graph $\Gamma=(V,E)$ is a set of vertices $S\subset V$ where for every vertex $v$ in its boundary it holds that the majority of vertices in $v$'s closed neighborhood are in $S$. In the case of strong offensive alliance, strict majority is required. An alliance $S$ is called global if it affects every vertex in $V\backslash S$, that is, $S$ is a dominating set of $\Gamma$. The global offensive alliance number $\gamma_o(\Gamma)$ (respectively, global strong offensive alliance number $\gamma_{\hat{o}}(\Gamma)$) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in $\Gamma$. If $\Gamma$ has global independent offensive alliances, then the \emph{global independent offensive alliance number} $\gamma_i(\Gamma)$ is the minimum cardinality among all independent global offensive alliances of $\Gamma$. In this paper we study mathematical properties of the global (strong) alliance number of cubic graphs. For instance, we show that for all connected cubic graph of order $n$, $$\frac{2n}{5}\le \gamma_i(\Gamma)\le \frac{n}{2}\le \gamma_{\hat{o}}(\Gamma)\le \frac{3n}{4} \le \gamma_{\hat{o}}({\cal L}(\Gamma))=\gamma_{o}({\cal L}(\Gamma))\le n,$$ where ${\cal L}(\Gamma)$ denotes the line graph of $\Gamma$. All the above bounds are tight

Global defensive k-alliances in graphs

Let $\Gamma=(V,E)$ be a simple graph. For a nonempty set $X\subseteq V$, and a vertex $v\in V$, $\delta_{X}(v)$ denotes the number of neighbors $v$ has in $X$. A nonempty set $S\subseteq V$ is a \emph{defensive $k$-alliance} in $\Gamma=(V,E)$ if $\delta_S(v)\ge \delta_{\bar{S}}(v)+k,$ $\forall v\in S.$ A defensive $k$-alliance $S$ is called \emph{global} if it forms a dominating set. The \emph{global defensive $k$-alliance number} of $\Gamma$, denoted by $\gamma_{k}^{a}(\Gamma)$, is the minimum cardinality of a defensive $k$-alliance in $\Gamma$. We study the mathematical properties of $\gamma_{k}^{a}(\Gamma)$

On defensive alliances and line graphs

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

On the Randi\'{c} index and conditional parameters of a graph

The aim of this paper is to study some parameters of simple graphs related with the degree of the vertices. So, our main tool is the $n\times n$ matrix ${\cal A}$ whose ($i,j$)-entry is $$a_{ij}= \left\lbrace \begin{array}{ll} \frac{1}{\sqrt{\delta_i\delta_j}} & {\rm if }\quad v_i\sim v_j ; \\ 0 & {\rm otherwise,} \end{array} \right.$$ where $\delta_i$ denotes the degree of the vertex $v_i$. We study the Randi\'{c} index and some interesting particular cases of conditional excess, conditional Wiener index, and conditional diameter. In particular, using the matrix ${\cal A}$ or its eigenvalues, we obtain tight bounds on the studied parameters.Comment: arXiv admin note: text overlap with arXiv:math/060243

Alliance free and alliance cover sets

A \emph{defensive} (\emph{offensive}) $k$-\emph{alliance} in $\Gamma=(V,E)$ is a set $S\subseteq V$ such that every $v$ in $S$ (in the boundary of $S$) has at least $k$ more neighbors in $S$ than it has in $V\setminus S$. A set $X\subseteq V$ is \emph{defensive} (\emph{offensive}) $k$-\emph{alliance free,} if for all defensive (offensive) $k$-alliance $S$, $S\setminus X\neq\emptyset$, i.e., $X$ does not contain any defensive (offensive) $k$-alliance as a subset. A set $Y \subseteq V$ is a \emph{defensive} (\emph{offensive}) $k$-\emph{alliance cover}, if for all defensive (offensive) $k$-alliance $S$, $S\cap Y\neq\emptyset$, i.e., $Y$ contains at least one vertex from each defensive (offensive) $k$-alliance of $\Gamma$. In this paper we show several mathematical properties of defensive (offensive) $k$-alliance free sets and defensive (offensive) $k$-alliance cover sets, including tight bounds on the cardinality of defensive (offensive) $k$-alliance free (cover) sets

Topological and spectral properties of random digraphs

We investigate some topological and spectral properties of Erd\H{o}s-R\'{e}nyi (ER) random digraphs $D(n,p)$. In terms of topological properties, our primary focus lies in analyzing the number of non-isolated vertices $V_x(D)$ as well as two vertex-degree-based topological indices: the Randi\'c index $R(D)$ and sum-connectivity index $\chi(D)$. First, by performing a scaling analysis we show that the average degree $\langle k \rangle$ serves as scaling parameter for the average values of $V_x(D)$, $R(D)$ and $\chi(D)$. Then, we also state expressions relating the number of arcs, spectral radius, and closed walks of length 2 to $(n,p)$, the parameters of ER random digraphs. Concerning spectral properties, we compute six different graph energies on $D(n,p)$. We start by validating $\langle k \rangle$ as the scaling parameter of the graph energies. Additionally, we reformulate a set of bounds previously reported in the literature for these energies as a function $(n,p)$. Finally, we phenomenologically state relations between energies that allow us to extend previously known bounds