The Shannon Capacity of Graph Powers

For a graph <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i>, its <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>-th graph power <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G^k</i> is constructed by placing an edge between two vertices if they are within distance <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>. We consider the problem of deriving upper bounds on the Shannon capacity of graph powers by using spectral graph theory and linear optimization methods. First, we use the so-called ratio-type bound to provide an alternative and spectral proof of a result by Lovász [<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEEE Trans. Inform. Theory</i> 1979], which states that, for a regular graph, the Hoffman ratio bound on the independence number is also an upper bound on the Lovász theta number and, hence, also on the Shannon capacity. In fact, we show that Lovász’ result holds in the more general context of graph powers. Secondly, we derive another bound on the Shannon capacity of graph powers, the so-called rank-type bound, which depends on a new family of polynomials that can be computed by running a simple algorithm. Lastly, we provide several computational experiments that demonstrate the sharpness of the two proposed algebraic bounds. As a by-product, when these two new algebraic bounds are tight, they can be used to easily derive the exact values of the Lovász theta number (which relies on solving an SDP) and the Shannon capacity (which is not known to be computable) of the corresponding graph power.