On weighted noncorona graphs with property R and −SR

Abstract

Let Gw be a simple weighted graph with adjacency matrix A(Gw). The set of all eigenvalues of A(Gw) is called the spectrum of weighted graph Gw denoted by σ(Gw). The reciprocal eigenvalue property (or property R) for a connected weighted nonsingular graph Gw is defined as, if η ∈ σ(Gw) then 1 η ∈ σ(Gw). Further, if η and 1 η have the same multiplicities for each η ∈ σ(Gw) then this graph is said to have strong reciprocal eigenvalue property (or property SR). Similarly, a connected weighted nonsingular graph Gw is said to have anti-reciprocal eigenvalue property (or property −R) if η ∈ σ(Gw) then −1 η ∈ σ(Gw). Furthermore, if η and −1 η have the same multiplicities for each η ∈ σ(Gw) then strong anti-reciprocal eigenvalue property (or property −SR) holds for the weighted graph Gw. In this article, classes of weighted noncorona graphs satisfying property R and property −SR are studied.