Background: An integral graph is a graph whose spectrum consists entirely of integers (see [1]).
Example: Complete graph $K_n$, since spectrum$(K_n) = (n-1,-1,\ldots,-1)$
Question Is the induced subgraph of integral graph is also integral?
In otherwords this means are the eigenvalues of induced subgraph of intgral graph are also integers.
Or is $ \lambda_i \in \mathbb{N}$ ? where $\lambda_i $ is any eigenvalue of induced subgraph $H$ of integral graph $G$.
Note this is true for the case of complete graph.
Idea We know that the eigenvalues of induced subgraph interlace with that of the graph (see Corollary 2.5.2 in [2] ). That would help us to determine some bounds for the eigenvalues of the induced subgraph.
Any help will be useful !!
This is not even close to true. The eigenvalues of the Petersen graph are integers, but it contains 5-cycles as induced subgraphs and these do not have integer eigenvalues. Or the eigenvalues of a 4-cycle are integers, but this contains paths on three vertices, which have non-integral eigenvalues. More generally note that the eigenvalues of the complete bipartite graph $K_{m,n}$ are integers if and only if $mn$ is a perfect square; this leads to many more examples.