ALGORITHMIC PROOF OF MaxMult(T) = p(T)

Title & Authors
ALGORITHMIC PROOF OF MaxMult(T) = p(T)
Kim, In-Jae;

Abstract
For a given graph G we consider a set S(G) of all symmetric matrices A = [$\small{a_{ij}}$] whose nonzero entries are placed according to the location of the edges of the graph, i.e., for $\small{i{\neq}j}$, $\small{a_{ij}{\neq}0}$ if and only if vertex $\small{i}$ is adjacent to vertex $\small{j}$. 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 $\small{n}$ - 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 $\small{p}$(T) of the tree T. This gives an alternative proof of the interesting result, MaxMult(T) = $\small{p}$(T).
Keywords
maximum corank;maximum multiplicity;minimum rank;path cover number;tree;
Language
English
Cited by
References
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