The spectrum of the hyper-star graphs and their line graphs

Let n 1 be an integer. The hypercube Qn is the graph whose vertex set isf0;1gn, where two n-tuples are adjacent if they differ in precisely one coordinate. This graph has many applications in Computer sciences and other area of sciences. Inthe graph Qn, the layer Lk is the set of vertices with exactly k 1’s, namely, vertices ofweight k, 1 k n. The hyper-star graph B(n;k) is the subgraph of Qn induced bylayers Lk and Lk+1; 0 < k < n. In this paper, we determine the spectrum of the hyperstargraph B(n;k) and L(B(n;k)), where L(B(n;k)) is the line graph of the graphB(n;k). In particular, we show that the graph L(B(n;k)) is an integral graph, that is,all of its eigenvalues are integers. In this paper, we investigate some of the algebraic properties of the graph B(n;k) andits line graph L(B(n;k)). In particular, we determine the spectrum of these graphs.