Constructing disjoint Steiner trees in Sierpi\'{n}ski graphs

Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$. Then the trees $T_1, T_2, \cdots, T_\ell$ in $G$ are \emph{internally disjoint Steiner trees} connecting $S$ (or $S$-Steiner trees) if $E(T_i) \cap E(T_j )=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for every pair of distinct integers $i,j$, $1 \leq i, j \leq \ell$. Similarly, if we only have the condition $E(T_i) \cap E(T_j )=\emptyset$ but without the condition $V(T_i)\cap V(T_j)=S$, then they are \emph{edge-disjoint Steiner trees}. The \emph{generalized $k$-connectivity}, denoted by $\kappa_k(G)$, of a graph $G$, is defined as $\kappa_k(G)=\min\{\kappa_G(S)|S \subseteq V(G) \ \textrm{and} \ |S|=k \}$, where $\kappa_G(S)$ is the maximum number of internally disjoint $S$-Steiner trees. The \emph{generalized local edge-connectivity} $\lambda_{G}(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $G$. The {\it generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is defined as $\lambda_k(G)=\min\{\lambda_{G}(S)\,|\,S\subseteq V(G) \ and \ |S|=k\}$. These measures are generalizations of the concepts of connectivity and edge-connectivity, and they and can be used as measures of vulnerability of networks. It is, in general, difficult to compute these generalized connectivities. However, there are precise results for some special classes of graphs. In this paper, we obtain the exact value of $\lambda_{k}(S(n,\ell))$ for $3\leq k\leq \ell^n$, and the exact value of $\kappa_{k}(S(n,\ell))$ for $3\leq k\leq \ell$, where $S(n, \ell)$ is the Sierpi\'{n}ski graphs with order $\ell^n$. As a direct consequence, these graphs provide additional interesting examples when $\lambda_{k}(S(n,\ell))=\kappa_{k}(S(n,\ell))$. We also study the some network properties of Sierpi\'{n}ski graphs

Urban Popular Economies

What is a life worth living and how is it concretely actualized by an urban majority making often unanticipated, unformatted uses of the urban to engender livelihoods in a dynamic and open-ended process? This is the key question undertaken in this collectively written piece. This means thinking about work, paid and unpaid, in ways that highlight the everyday practices of urban inhabitants as they put together territories in which to operate, which sustain their imaginations of well-being as part of a process of being with others—in households, neighborhoods, communities, and institutions. What is it that different kinds of workers have in common; what links them; where does the household begin and end; what is the difference between productive and reproductive work