On graphs with strong anti-reciprocal eigenvalue property

Abstract

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 (Formula presented.) is an eigenvalue of A(G) whenever ? is an eigenvalue of A(G). Further, if ? and (Formula presented.) have the same multiplicity for each eigenvalue ?, then it is said to have the strong anti-reciprocal eigenvalue property (property (?SR)). A graph G is called unimodular if (Formula presented.). 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). � 2021 Informa UK Limited, trading as Taylor & Francis Group.