Induced subgraph and eigenvalues of some signed graphs

Abstract

Hao Huang proved the Sensitivity Conjecture in [Induced graphs of the hypercube and a proof of the Sensitivity Conjecture, Annals of Mathematics, 190 (2019), 949-955] by signed graph spectral method. The main result of that paper is every -vertex induced subgraph of n-dimensional hypercube Qn has maximum degree at least by proving that Qn has just two distinct adjacency eigenvalues and with a signature. In this paper, we consider the eigenvalues of signed Cartesian product of bipartite graph and hypercube Qn , signed Cartesian product of complete graph Km and hypercube Qn which contain Huang’s result when or m = 2. Let f(G) be the minimum of the maximum degree of an induced subgraph of G on vertices where is the independent number of G. Huang also asked the following question: Given a graph G with high symmetry, what can we say about f(G)? We study this question for k-ary n-cubes where 2-ary n-cube is Qn . In this paper, we show that every -vertex induced subgraph of has maximum degree at least () by proving that has two eigenvalues which are either or with a signature.