The a-average Degree Edge-Connectivity of Bijective Connection Networks.

The conditional edge-connectivity is an important parameter to evaluate the reliability and fault tolerance of multi-processor systems. The |$n$| -dimensional bijective connection networks |$B_{n}$| contain hypercubes, crossed cubes, Möbius cubes and twisted cubes, etc. The conditional edge-connectivity of a connected graph |$G$| is the minimum cardinality of edge sets, whose deletion disconnects |$G$| and results in each remaining component satisfying property |$\mathscr{P}$|⁠. And let |$F$| be the edge set as desired. For a positive integer |$a$|⁠ , if |$\mathscr{P}$| denotes the property that the average degree of each component of |$G-F$| is no less than |$a$|⁠ , then the conditional edge-connectivity can be called the |$a$| -average degree edge-connectivity |$\overline{\lambda }_{a}(G)$|⁠. In this paper, we determine that the exact value of the |$a$| -average degree edge-connectivity of an |$n$| -dimensional bijective connection network |$\overline{\lambda }_{a}(B_{n})$| is |$(n-a)2^a$| for each |$0\leq a \leq n-1 $| and |$n\geq 1$|⁠. ¹ [ABSTRACT FROM AUTHOR]