Unbalanced signed graphs with eigenvalue properties

For a signature function $\mathrm{\Psi }:E(H)⟶\{\pm 1\}$ with underlying graph $H$, a signed graph (S.G) $\hat{H}=(H,\mathrm{\Psi })$ is a graph in which edges are assigned the signs using the signature function $\mathrm{\Psi }$. An S.G $\hat{H}$ is said to fulfill the symmetric eigenvalue property if for every eigenvalue $\hat{h}(\hat{H})$ of $\hat{H}$, $-\hat{h}(\hat{H})$ is also an eigenvalue of $\hat{H}$. A non singular S.G $\hat{H}$ is said to fulfill the property $(\mathcal{S}\mathcal{R})$ if for every eigenvalue $\hat{h}(\hat{H})$ of $\hat{H}$, its reciprocal is also an eigenvalue of $\hat{H}$ (with multiplicity as that of $\hat{h}(\hat{H})$). A non singular S.G $\hat{H}$ is said to fulfill the property $(-\mathcal{S}\mathcal{R})$ if for every eigenvalue $\hat{h}(\hat{H})$ of $\hat{H}$, its negative reciprocal is also an eigenvalue of $\hat{H}$ (with multiplicity as that of $\hat{h}(\hat{H})$). In this article, non bipartite unbalanced S.Gs ${\hat{\mathfrak{C}}}_3^{(m,1)}$ and ${\hat{\mathfrak{C}}}_5^{(m,2)}$, where $m$ is even positive integer have been constructed and it has been shown that these graphs fulfill the symmetric eigenvalue property, the S.Gs ${\hat{\mathfrak{C}}}_3^{(m,1)}$ also fulfill the properties $(-\mathcal{S}\mathcal{R})$ and $(\mathcal{S}\mathcal{R})$, whereas the S.Gs ${\hat{\mathfrak{C}}}_5^{(m,2)}$ are close to fulfill the properties $(-\mathcal{S}\mathcal{R})$ and $(\mathcal{S}\mathcal{R})$.