A concentration phenomenon for $h$-extra edge-connectivity reliability analysis of enhanced hypercubes $Q_{n,2}$ with exponentially many faulty links

Reliability assessment of interconnection networks is critical to the design and maintenance of multiprocessor systems. The $(n, k)$-enhanced hypercube $Q_{n,k}$, as a variation of the hypercube $Q_{n}$, was proposed by Tzeng and Wei in 1991. As an extension of traditional edge-connectivity, $h$-extra edge-connectivity of a connected graph $G,$ $\lambda_h(G),$ is an essential parameter for evaluating the reliability of interconnection networks. This article intends to study the $h$-extra edge-connectivity of the $(n,2)$-enhanced hypercube $Q_{n,2}$. Suppose that the link malfunction of an interconnection network $Q_{n,2}$ does not isolate any subnetwork with no more than $h-1$ processors, the minimum number of these possible faulty links concentrates on a constant $2^{n-1}$ for each integer $\lceil\frac{11\times2^{n-1}}{48}\rceil \leq h \leq 2^{n-1}$ and $n\geq 9$. That is, for about $77.083\%$ of values where $h\leq2^{n-1},$ the corresponding $h$-extra edge-connectivity of $Q_{n,2}$, $\lambda_h(Q_{n,2})$, presents a concentration phenomenon. Moreover, the lower and upper bounds of $h$ mentioned above are both tight.