This paper considers the edge-connectivity and restricted edge-connectivityof replacement product graphs, gives some bounds on edge-connectivity andrestricted edge-connectivity of replacement product graphs and determines theexact values for some special graphs. In particular, the authors furtherconfirm that under certain conditions, the replacement product of two Cayleygraphs is also a Cayley graph, and give a necessary and sufficient conditionfor such Cayley graphs to have maximum restricted edge-connectivity. Based onthese results, the authors construct a Cayley graph with degree $d$ whoserestricted edge-connectivity is equal to $d+s$ for given odd integer $d$ andinteger $s$ with $d \geqslant 5$ and $1\leqslant s\leqslant d-3$, which answersa problem proposed ten years ago.
展开▼
机译：本文考虑了替换乘积图的边连接性和限制边连接性，给出了替换乘积图的边连接性和限制边连接性的界限，并确定了某些特殊图的精确值。特别是，作者进一步确认，在某些条件下，两个Cayley图的替换乘积也是Cayley图，并为此类Cayley图具有最大的受限边连接性提供了必要和充分的条件。基于这些结果，作者构建了一个度数为d $的Cayley图，对于给定的奇数整数dd $和整数$ s $，其限制的边缘连接性等于$ d + s $，其中$ d \ geqslant 5 $和$ 1 \ leqslant s \ leqslant d-3 $，它回答了十年前提出的一个问题。
展开▼
