Maximum nullity of outerplanar graphs and the path cover number |
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Authors: | John Sinkovic |
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Institution: | Department of Mathematics, Brigham Young University, Provo, UT 84602, United States |
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Abstract: | Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=aij] with aij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte 5], have shown that for a tree T,M(T)=P(T). Barioli et al. 2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G). |
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Keywords: | 05C50 05C10 15A18 15A03 15B57 |
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