On multiple eigenvalues of trees

Citation

Rowlinson P (2010) On multiple eigenvalues of trees. Linear Algebra and Its Applications, 432 (11), pp. 3007-3011. https://doi.org/10.1016/j.laa.2010.01.003

Abstract
Let T be a tree of order n>6 with μ as a positive eigenvalue of multiplicity k. Star complements are used to show that (i) if k>n/3 then μ=1, (ii) if μ=1 then, without restriction on k, T has k+1 pendant edges that form an induced matching. The results are used to identify the trees with a non-zero eigenvalue of maximum possible multiplicity.

Keywords
Graph; Tree; Eigenvalue; Star complement

Journal
Linear Algebra and Its Applications: Volume 432, Issue 11