Computation of minimal rank and path cover number for certain graphs
نویسندگان
چکیده
For a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is non-zero whenever i / =j and {i, j} is an edge in G. The path cover number of G is the minimum number of vertexdisjoint paths occurring as induced subgraphs of G that cover all the vertices of G. For trees, the relationship between minimum rank and path cover number is completely understood. However, for non-trees only sporadic results are known. We derive formulae for the minimum rank and path cover number for graphs obtained from edge-sums, and formulae for minimum rank of vertex sums of graphs. In addition we examine previously identified special types of vertices and attempt to unify the theory behind them. © 2004 Elsevier Inc. All rights reserved. AMS classification: 15A18; 05C50
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