On graphs with anti-reciprocal eigenvalue property

Let \mathtt{A}(\mathtt{G}) be the adjacency matrix of a simple connected undirected graph \mathtt{G}. A graph \mathtt{G} of order n is said to be non-singular (respectively singular) if \mathtt{A}(\mathtt{G}) is non-singular (respectively singular). The spectrum of a graph \mathtt{G} is the set of all its eigenvalues denoted by spec(\mathtt{G}). The anti-reciprocal (respectively reciprocal) eigenvalue property for a graph \mathtt{G} can be defined as `` Let \mathtt{G} be a non-singular graph \mathtt{G} if the negative reciprocal (respectively positive reciprocal) of each eigenvalue is likewise an eigenvalue of \mathtt{G}, then \mathtt{G} has anti-reciprocal (respectively reciprocal) eigenvalue property ." Furthermore, a graph \mathtt{G} is said to have strong anti-reciprocal eigenvalue property (resp. strong reciprocal eigenvalue property) if the eigenvalues and their negative (resp. positive) reciprocals are of same multiplicities. In this article, graphs satisfying anti-reciprocal eigenvalue (or property (-\mathtt{R})) and strong anti-reciprocal eigenvalue property (or property (-\mathtt{SR})) are discussed.

Keywords:

Anti-reciprocal eigenvalue property , strong anti-reciprocal eigenvalue property , Adjacency Matrix , graph spectrum