Determination of (0,2)-regular sets in graphs - Pacheco, Maria F. (2014)
An eigenvalue of a graph is main iff its associated eigenspace is not orthogonal to the all-one vector j. The main characteristic polynomial of a graph G with p main distinct eigenvalues is _ (λ)=λ^−_0 λ^(−1)−_1 λ^(−2)-…-_(−2) λ−_(−1) and it has integer coefficients. If G has n vertices, the nxk walk matrix of G is _=(j,_j,_^"2" "j",…,_^(−) j) and W, the walk matrix of G, is _ for which rank(_)=k. The number k coincides with the number of distinct main eigenvalues of G. In [2] it was proved that the coefficients of the main characteristic polynomial of G are the solutions of =_^j. A (,)- regular set [3] is a subset of the vertices of a graph inducing a -regular subgraph such that every vertex not in the subset has  neighbors in it. In [1], a strategy for the determination of (0,1)-regular sets is described and we generalize it in order to solve the problem of the determination of (0,2)-regular sets in arbitrary graphs. An algorithm for deciding whether or not a given graph has a (0,2)-regular set is described. Its complexity depends on the multiplicity of −2 as an eigenvalue of the adjacency matrix of the graph. When such multiplicity is low, the generalization of the results in [1] assure that the algorithm is polynomial. An example of application of the algorithm to a graph for which this multiplicity is low is also presented.