79-89 G UNICYCLING GRAPHS WITH THE STRONG RECIPROCAL EIGENVALUE PROPERTY

GRAPH UNISIKLIK DENGAN SIFAT-SIFAT EIGENVALUE RESIPROK KUAT

Abstract

A graph G is bipartite if and only if the negative of each eigenvalue of G is also an
eigenvalue of G. It is said that a graph has property (R), if G is nonsingular and the
reciprocal of each of its eigenvalues is also an eigenvalue. Further, if the multiplicity of an
eigenvalue equals that of its reciprocal, the graph is said to have property (SR). The trees
with property (SR) have been recently characterized by Barik, Pati and Sarma. Barik,
Neumann and Pati have shown that for trees the two properties are, in fact, equivalent. In
this paper, the structure of a unicyclic graph with property (SR) is studied. It has been shown
that such a graph is bipartite and is a corona (unless it has girth four). In the case it is not a
corona, it is shown that the graph can have one of the three specified structures. Families of
unicyclic graphs with property (SR) having each of these specific structures are provided.
Key words. Unicyclic graphs, Adjacency matrix, Corona, Perfect matching, Property (SR).