Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The k-thpower of a graph G=(V,E), G^k, is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. This article proves various eigenvalue bounds for the independence number and chromatic number of G^k which purely depend on the spectrum of G, together with a method to optimize them. Our bounds for the k-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well.The research of A. Abiad is partially supported by the FWO grant 1285921N. A. Abiad and M.A. Fiol gratefully acknowledge the support from DIAMANT. This research of M.A. Fiol has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. B. Nogueira acknowledges grant PRPQ/ADRC from UFMG. The authors would also like to thank Anurag Bishnoi for noticing a tight family for our bound (19).Peer ReviewedPostprint (author's final draft

Multidiameters and Multiplicities

AbstractThe k -diameter of a graph Γ is the largest pairwise minimum distance of a set ofk vertices in Γ, i.e., the best possible distance of a code of size k in Γ. Ak -diameter for some k is called a multidiameter of the graph. We study the function N(k,Δ , D), the largest size of a graph of degree at most Δ and k -diameter D. The graphical analogues of the Gilbert bound and the Hamming bound in coding theory are derived. Constructions of large graphs with given degree and k -diameter are given. Eigenvalue upper bounds are obtained. By combining sphere packing arguments and eigenvalue bounds, new lower bounds on spectral multiplicity are derived. A bound on the error coefficient of a binary code is given