Let \(n >3\) and \( 0< k < \frac{n}{2} \) be integers. In this paper, we investigate some algebraic properties of the line graph of the graph \( {Q_n}(k,k+1) \) where \( {Q_n}(k,k+1) \) is the subgraph of the hypercube \(Q_n\) which is induced by the set of vertices of weights k and \(k+1\). The graph \( {Q_n}(k,k+1) \) has a close relation to Johnson graph \(J(n+1,k+1)\). In fact, it is the square root of the graph \(J(n+1,k+1)\). We will see that when \( n\ne 2k+1\), then the graph \( {Q_n}(k,k+1) \) is a non-regular edge-transitive graph; hence, its line graph is a vertex-transitive graph. In the first step, we determine the automorphism groups of these graphs for all values of n, k. In the second step, we study Cayley properties of the line graphs of these graphs. In particular, we show that if \(k\ge 3\) and \( n \ne 2k+1\), then except for the cases \((k,n) \ne (3,9)\) and \((k,n) \ne (3,33)\), the line graph of the graph \( {Q_n}(k,k+1) \) is a vertex-transitive non-Cayley graph. Also, we show that the line graph of the graph \( {Q_n}(1,2) \) is a Cayley graph if and only if n is a power of a prime p. Moreover, we show that for ‘almost all’ even values of k, the line graph of the graph \( {Q_{2k+1}}(k,k+1) \) is a vertex-transitive non-Cayley graph.

令\（n> 3 \）和\（0 <k <\ frac {n} {2} \）为整数。在本文中，我们研究了图\（{Q_n}（k，k + 1）\）的线图的代数性质，其中\（{Q_n}（k，k + 1）\）是图的子图超立方体\（Q_n \）由权重k和\（k + 1 \）的顶点集合诱导。图\（{Q_n}（k，k + 1）\）与Johnson图\（J（n + 1，k + 1）\）有密切关系。实际上，它是图\（J（n + 1，k + 1）\）的平方根。我们将看到，当\（n \ ne 2k + 1 \）时，图\（{Q_n}（k，k + 1）\）是不规则的边缘传递图；因此，它的线图是顶点传递图。第一步，我们确定所有n，  k值的这些图的自同构群。在第二步中，我们研究这些图的折线图的Cayley属性。特别地，我们表明如果\（k \ ge 3 \）和\（n \ ne 2k + 1 \），则除了\（（k，n）\ ne（3,9）\）和\ （（k，n）\ ne（3,33）\），图\（{Q_n}（k，k + 1）\）的折线图是顶点传递的非Cayley图。此外，我们证明了当（仅当）n时，图\（{Q_n}（1,2）\）的线图是Cayley图。是素数p的幂。此外，我们表明，对于k的“几乎所有”偶数值，图\（{Q_ {2k + 1}}（k，k + 1）\）的折线图是可传递顶点的非Cayley图。