The Shannon capacity of graph powers

Abstract

For a graph G, its k-th graph power Gk is constructed by placing an edge between two vertices if they are within distance k. 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 Lovasz [ IEEE Trans. Inform. Theory 1979], which states that, for a regular graph, the Hoffman ratio bound on the independence number is also an upper bound on the Lovasz theta number and, hence, also on the Shannon capacity. In fact, we show that Lovasz’ 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 xperiments 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 Lovasz theta number which relies on solving an SDP) and the Shannon capacity (which is not known to be computable) of the corresponding graph power.

Abiad is supported by NWO (Dutch Research Council) through the grants VI.Vidi.213.085 and OCENW.KLEIN.475. C. Dalfo and M. A. Fiol are funded by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RB-I00. M. A. Fiol’s research is also supported by a grant from the Universitat Politecnica de Catalunya, reference AGRUPS-2025. The authors thank Luuk Reijnders for his support with Sagemath. The first author thanks Jeroen Zuiddam for inspiring discussions on the topic.