Let G be the Cayley graph of the elementary abelian 2-group ℤn2 with respect to a set S of size d. We prove that for any such G,S and d, the maximum degree of any induced subgraph of G on any set of more than half the vertices is at least √d. This is deduced from the recent breakthrough result of Huang who proved the above for the n-hypercube Qn, in which the set of generators S is the set of all vectors of Hamming weight 1. Motivated by his method we define and study unitary signings of adjacency matrices of graphs, and compare them to the orthogonal signings of Huang. As a byproduct, we answer a recent question of Belardo, Cioabă, Koolen, and Wang about the spectrum of signed 5-regular graphs.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics