ALGORITHMIC PROOF OF MaxMult(T) = p(T)
In‐Jae Kim · Communications of the Korean Mathematical Society · 2012
For a given graph G we consider a set S(G) of all symmetric matrices A = [ $a_{ij}$ ] whose nonzero entries are placed according to the location of the edges of the graph, i.e., for $i{ eq}j$ , $a_{ij}{ eq}0$ if and only if vertex $i$ is adjacent to vertex $j$ . The minimum rank mr(G) of the graph G is defined to be the smallest rank of a matrix in S(G). In general the computation of mr(G) is complicated, and so is that of the maximum multiplicity MaxMult(G) of an eigenvalue of a matrix in S(G) which is equal to $n$ - mr(G) where n is the number of vertices in G. However, for trees T, there is a recursive formula to compute MaxMult(T). In this note we show that this recursive formula for MaxMult(T) also computes the path cover number $p$ (T) of the tree T. This gives an alternative proof of the interesting result, MaxMult(T) = $p$ (T).