Let T be a tree, we show that the null space of the adjacency matrix of T has relevant information about the structure of T . We introduce the Null Decomposition of trees, and use it in order to get formulas for independence number and matching number of a tree. We also prove that the number of maximum matchings in a tree is related to the null decomposition.
We present a necessary and sufficient condition for the singularity of circulant matrices associated with directed weighted cycles. This condition is simple and independent of the order of matrices from a complexity point of view. We give explicit and simple formulas for the Drazin inverse of these circulant matrices. We also provide a Bjerhammar-type condition for the Drazin inverse.
We characterize unicyclic graphs that are singular using the support of the null space of their pendant trees. From this, we obtain closed formulas for the independence and matching numbers of a unicyclic graph, based on the support of its subtrees. These formulas allows one to compute independence and matching numbers of unicyclic graphs using linear algebra methods.1991 Mathematics Subject Classification. 05C50, 15A18.
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