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.