Let G be a simple connected graph and A(G) be the adjacency matrix of G. Then G is said to have the anti-reciprocal eigenvalue property (property (−R)) if $-\frac{1}{\lambda }$ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and $-\frac{1}{\lambda }$ have the same multiplicity for each eigenvalue λ, then it is said to have the strong anti-reciprocal eigenvalue property (property (−SR)). A graph G is called unimodular if $det\left(A\right(G\left)\right)=\pm 1$. We prove that if G satisfies property (−R), then G is unimodular. Further, we provide a complete characterization of all non-bipartite unicyclic graphs satisfying property (−SR) and prove that those graphs are necessarily coronas. In Ahmad et al. [Noncorona graphs with strong anti-reciprocal eigenvalue property. Linear Multilinear Alg. 2021;69(10):1878–1888], the authors constructed few classes of noncorona graphs using corona triangles, corona squares and corona pentagons which satisfy property (−SR). As a concluding remark, the authors suggested the following: ‘property (−SR) seems to be hold for any class constructed in the same way using corona cycles of any finite length’. We prove it to be true and construct a more general class of noncorona graphs satisfying property (−SR).

令G为简单连通图，A ( G ) 为G的邻接矩阵。则称G具有反互反特征值属性（属性 (−R)）$-\frac{\mathrm{1个}}{\lambda }$是A ( G )的特征值，只要λ是A ( G )的特征值。此外，如果λ和$-\frac{\mathrm{1个}}{\lambda }$对每个特征值λ具有相同的重数，则称其具有强反互反特征值性质（性质 (−SR)）。如果图G称为幺模图$检测\left(A\right(G\left)\right)=\pm \mathrm{1个}$. 我们证明如果G满足属性 (−R)，则G是幺模的。此外，我们提供了满足属性 (−SR) 的所有非二部单环图的完整表征，并证明这些图必然是冠状图。在艾哈迈德等人。[Noncorona graphs with strong anti-reciprocal eigenvalue property。线性多线性算法。2021;69(10):1878–1888]，作者使用满足属性 (-SR) 的电晕三角形、电晕正方形和电晕五边形构建了几类非电晕图。作为结束语，作者提出以下建议：“属性 (−SR) 似乎适用于使用任何有限长度的电晕循环以相同方式构建的任何类”。我们证明它是正确的，并构建了一个更通用的非电晕图类，满足属性 (−SR)。