Strong reciprocal eigenvalue property of a class of weighted graphs

Abstract

Let H be the class of connected bipartite graphs G with a unique perfect matching M. For G∈H, let WG be the set of weight functions w on the edge set E(G) such that w(e)=1 for each matching edge and w(e)>0 for each nonmatching edge. Let Gw denote the weighted graph with G∈H and w∈WG. The graph Gw is said to satisfy the reciprocal eigenvalue property, property (R), if 1/λ is an eigenvalue of the adjacency matrix A(Gw) whenever λ is an eigenvalue of A(Gw). Moreover, if the multiplicities of the reciprocal eigenvalues are the same, we say Gw has the strong reciprocal eigenvalue property, property (SR). Let Hg={G∈H|G/M is bipartite}, where G/M is the graph obtained from G by contracting each edge in M to a vertex.Recently in [12], it was shown that if G∈Hg, then Gw has property (SR) for some w∈WG if and only if Gw has property (SR) for each w∈WG if and only if G is a corona graph (obtained from another graph H by adding a new pendant vertex to each vertex of H).Now we have the following questions. Is there a graph G∈H∖Hg such that Gw has property (SR) for each w∈WG? Are there graphs G∈H∖Hg such that Gw never has property (SR), not even for one w∈WG? Are there graphs G∈H such that Gw has property (SR) for some w∈WG but not for all w∈WG? In this article, we supply answers to these three questions. We also supply a graph class larger than Hg where for any graph G, if Gw has property (SR) for one w∈WG, then G is a corona graph.