Spectral Characterization of Graphs with Respect to the Anti-Reciprocal Eigenvalue Property

Abstract

Let G=(V,E) be a simple connected graph with vertex set V and edge set E, respectively. The term “anti-reciprocal eigenvalue property“ refers to a non-singular graph G for which, −1λ∈σ(G), whenever λ∈σ(G), ∀λ∈σ(G). Here, σ(G) is the multiset of all eigenvalues of A(G). Moreover, if multiplicities of eigenvalues and their negative reciprocals are equal, then that graph is said to have strong anti-reciprocal eigenvalue properties, and the graph is referred to as a strong anti-reciprocal graph (or (−SR) graph). In this article, a new family of graphs Fn(k,j) is introduced and the energy of F5(k,k2)k≥2 is calculated. Furthermore, with the help of F5(k,k2), some families of (−SR) graphs are constructed.