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Determination of (0,2)-regular sets in graphs
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Advisor(s)
Abstract(s)
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 𝑚_𝐺 (λ)=λ^𝑝−𝑐_0 λ^(𝑝−1)−𝑐_1 λ^(𝑝−2)-…-𝑐_(𝑝−2) λ−𝑐_(𝑝−1) and it has integer coefficients. If G has n vertices, the nxk walk matrix of G is 𝑾_𝒌=(j,𝑨_𝑮j,𝑨_𝑮^"2" "j",…,𝑨_𝑮^(𝒌−𝟏) j) and W, the walk matrix of G, is 𝑾_𝒌 for which rank(𝑾_𝒌)=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 𝑾𝑿=𝑨_𝑮^𝒑j. 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.
Description
Keywords
(0,2)-Regular sets Main eigenvalues Maximum matching
Citation
Pacheco, Maria F.; Cardoso, Domingos Moreira; Luz, Carlos J. (2013). Determination of (0,2)-regular sets in graphs. In Research Day - Universidade de Aveiro. Aveiro
Publisher
Universidade de Aveiro