Let
H be the class of connected bipartite graphs
G with a unique perfect matching
6c0ecf86f2afff192e94e" title="Click to view the MathML source">M. For
46d3fb7790bbd0ae685610fde090d7" title="Click to view the MathML source">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
46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw denote the weighted graph with
46d3fb7790bbd0ae685610fde090d7" title="Click to view the MathML source">G∈H and
w∈WG. The graph
46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">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
46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has the strong reciprocal eigenvalue property,
property (SR) . Let
6c706137961bf352239bf60f0" title="Click to view the MathML source">Hg={G∈H|G/M is bipartite}, where
G/M is the graph obtained from
G by contracting each edge in
6c0ecf86f2afff192e94e" title="Click to view the MathML source">M to a vertex.
Recently in [12], it was shown that if G∈Hg, then 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has property (SR) for some w∈WG if and only if 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">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 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has property (SR) for each w∈WG? Are there graphs G∈H∖Hg such that 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw never has property (SR), not even for one w∈WG? Are there graphs 46d3fb7790bbd0ae685610fde090d7" title="Click to view the MathML source">G∈H such that 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">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 6c3a95935bc9b2de89" title="Click to view the MathML source">Hg where for any graph G , if 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has property (SR) for one w∈WG, then G is a corona graph.