Some New Families of Noncorona Graphs with Strong Anti-Reciprocal Eigenvalue Property

Let G be a simple graph and A(G) be its adjacency matrix. We say that the graph G satisfies reciprocal eigenvalue property (or simply, property (R)) if 1λ is an eigenvalue of A(G), whenever λ is an eigenvalue of A(G). The graph G satisfies strong anti-reciprocal property (or simply, property (-SR)) if -1λ is an eigenvalue of A(G) with multiplicity k, whenever λ is an eigenvalue of A(G) with multiplicity k. Recently, Barik, Mondal and Pati proved that there exists no singular (non-trivial) tree whose nonzero eigenvalues satisfy reciprocal eigenvalue property. As a concluding remark, the authors mentioned that “It is interesting to see that our result (for trees) is true for any connected graph”. In this paper, we prove that there exists no singular unicyclic graph whose nonzero eigenvalues satisfy reciprocal eigenvalue property. Ahmad, Hameed and Jabeen in their recent paper raised the question about existence of noncorona graphs with property (-SR) and constructed seven classes of graphs satisfying this property. Here, we give several families of noncorona graphs with property (-SR).