From Graph Isoperimetric Inequality to Network Connectivity -- A New Approach

We present a new, novel approach to obtaining a network's connectivity. More specifically, we show that there exists a relationship between a network's graph isoperimetric properties and its conditional connectivity. A network's connectivity is the minimum number of nodes, whose removal will cause the network disconnected. It is a basic and important measure for the network's reliability, hence its overall robustness. Several conditional connectivities have been proposed in the past for the purpose of accurately reflecting various realistic network situations, with extra connectivity being one such conditional connectivity. In this paper, we will use isoperimetric properties of the hypercube network to obtain its extra connectivity. The result of the paper for the first time establishes a relationship between the age-old isoperimetric problem and network connectivity.Comment: 17 pages, 0 figure

Vulnerability of super edge-connected graphs

A subset $F$ of edges in a connected graph $G$ is a $h$-extra edge-cut if $G-F$ is disconnected and every component has more than $h$ vertices. The $h$-extra edge-connectivity \la^{(h)}(G) of $G$ is defined as the minimum cardinality over all $h$-extra edge-cuts of $G$. A graph $G$, if \la^{(h)}(G) exists, is super-\la^{(h)} if every minimum $h$-extra edge-cut of $G$ isolates at least one connected subgraph of order $h+1$. The persistence $\rho^{(h)}(G)$ of a super-\la^{(h)} graph $G$ is the maximum integer $m$ for which $G-F$ is still super-\la^{(h)} for any set $F\subseteq E(G)$ with $|F|\leqslant m$. Hong {\it et al.} [Discrete Appl. Math. 160 (2012), 579-587] showed that \min\{\la^{(1)}(G)-\delta(G)-1,\delta(G)-1\}\leqslant \rho^{(0)}(G)\leqslant \delta(G)-1, where $\delta(G)$ is the minimum vertex-degree of $G$. This paper shows that \min\{\la^{(2)}(G)-\xi(G)-1,\delta(G)-1\}\leqslant \rho^{(1)}(G)\leqslant \delta(G)-1, where $\xi(G)$ is the minimum edge-degree of $G$. In particular, for a $k$-regular super-\la' graph $G$, $\rho^{(1)}(G)=k-1$ if \la^{(2)}(G) does not exist or $G$ is super-\la^{(2)} and triangle-free, from which the exact values of $\rho^{(1)}(G)$ are determined for some well-known networks