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On edge star sets in trees

Akbari, S ; Sharif University of Technology | 2011

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2010.06.016
  3. Publisher: 2011
  4. Abstract:
  5. Let A be a Hermitian matrix whose graph is G (i.e. there is an edge between the vertices i and j in G if and only if the (i,j) entry of A is non-zero). Let λ be an eigenvalue of A with multiplicity mA(λ). An edge e=ij is said to be Parter (resp., neutral, downer) for λ,A if mA(λ)-mA-e(λ) is negative (resp., 0, positive ), where A-e is the matrix resulting from making the (i,j) and (j,i) entries of A zero. For a tree T with adjacency matrix A a subset S of the edge set of G is called an edge star set for an eigenvalue λ of A, if |S|=mA(λ) and A-S has no eigenvalue λ. In this paper the existence of downer edges and edge star sets for non-zero eigenvalues of the adjacency matrix of a tree is proved. We prove that neutral edges always exist for eigenvalues of multiplicity more than 1. It is also proved that an edge e=uv is a downer edge for λ,A if and only if u and v are both downer vertices for λ,A; and e=uv is a neutral edge if u and v are neutral vertices. Among other results, it is shown that any edge star set for each eigenvalue of a tree is a matching
  6. Keywords:
  7. Edge star set ; Eigenvalues ; Star set ; Trees
  8. Source: Discrete Mathematics ; Volume 311, Issue 13 , July , 2011 , Pages 1172-1178 ; 0012365X (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0012365X10002219